Cohomological Invariants of Homogeneous Varietieskarpenko/publ/hbl.pdf · Galois cohomology group H3(F(X)/F,Z/2), where F(X) is the function ﬁeld of X. We compute these groups for

Cohomological Invariants

of Homogeneous Varieties(with Applications to Quadratic Forms

and Central Simple Algebras)

Habilitationsschrift

Reine Mathematik

Fachbereich Mathematik

der Westfalischen Wilhelms-Universitat Munster

vorgelegt von Nikita Karpenko

aus St. Petersburg

3

Preface

Let X be a projective homogeneous variety over a field F . We study the Chowgroup CH∗(X) of algebraic cycles on X modulo rational equivalence gradedby codimension of cycles. Especially, we are interested in the Chow groupCH2(X) of 2-codimensional cycles because of its connection with the relativeGalois cohomology group H3(F (X)/F,Z/2), where F (X) is the function fieldof X. We compute these groups for various types of X and apply the resultsto problems in the theories of quadratic forms (e.g. isotropy over functionfields of homogeneous varieties) and central simple algebras (decomposability,common splitting fields).

Each chapter is written as an independent article with an abstract and anintroduction.

Results of Chapters 3–7 are obtained in joint work with Oleg Izhboldin.

Support of the Deutsche Forschungsgemeinschaft (Habilitatndenstipendium KA 1234/1-1) is gratefully acknowledged.

4

Contents

Chapter 1. Codimension 2 cycles on Severi-Brauer varieties 90. Introduction 91. Reduction to the primary case 112. Chern classes and gamma-filtration 123. Main observation 174. Computation of gamma-filtration 205. Algebras of prime exponent 306. Appendix 32

Chapter 2. Codimension 2 cycles on products of Severi-Brauer varieties 350. Introduction 351. Two preliminary results 362. Grothendieck group of product of Severi-Brauer varieties 363. Disjoint varieties and disjoint algebras 374. “Generic” varieties 395. “Generic” algebras 406. Biquaternion variety times conic 427. Product of two Severi-Brauer surfaces 45

Chapter 3. Isotropy of virtual Albert forms over function fields of quadrics 490. Introduction 491. Terminology, notation, and backgrounds 502. Computation of H3(F (SB(A)× SB(B))/F ) 523. Computation of H3(F (Xψ × SB(D))/F ) 544. 8-dimensional quadratic forms 565. Main theorem 57

Chapter 4. Isotropy of 6-dimensional quadratic forms over function fieldsof quadrics 59

0. Introduction 591. Terminology, notation, and backgrounds 612. The group H3(F (ρ1, ρ2)/F ) 623. The Grothendieck group of a quadric 654. The Grothendieck group of a product of quadrics 685. CH2 of a product of quadrics 696. The group I3(F (ρ, ψ)/F ) 747. The case of index 1 75

5

6 CONTENTS

8. Main theorem 789. The case of index 2 81

Chapter 5. Some new examples in the theory of quadratic forms 830. Introduction 831. Terminology, notation, and backgrounds 852. Products of Severi-Brauer varieties 863. The Grothendieck group of a grassmanian 884. Pull-back to generic fiber 925. Weil transfer via Galois descent 936. Galois action on Grothendieck group 957. Product of conics 968. Preliminary calculations I 989. Preliminary calculations II 10310. First basic construction 10911. Second basic construction 11112. Quadratic forms over complete fields 11313. 8-dimensional quadratic forms ϕ ∈ I2(F ) 11514. 14-dimensional quadratic forms ϕ ∈ I3(F ) 11915. Nonstandard isotropy 121

Chapter 6. On the group H3(F (ψ,D)/F ) 1230. Introduction 1231. Terminology, notation, and backgrounds 1242. The group TorsG∗K(X) 1253. Auxiliary lemmas 1264. On the group CH∗(X × Y ) 1285. The group TorsCH2(Xψ ×XD) 1296. The group H3(F (ψ,D)/F ) 1317. Corollaries 1338. A generalization 135

Chapter 7. Isotropy of 8-dimensional quadratic forms over function fieldsof quadrics 137

0. Introduction 1371. Terminology and notation 1382. The groups H3(F (ψ,D)/F ) and I3(F (ψ,D)/F ) 1393. The group K(Xψ ×XD) 1414. The group TorsCH2(Xψ ×XD) 1425. Standard isotropy in the case TorsCH2(Xψ ×XD) = 0 1456. The group H3(F (ψ,D)/F ) in the case ind(C0(ψ)⊗F D) = 2 1467. Main theorem 1498. Isotropy over the function field of a conic 151

Chapter 8. A generalization of the Albert-Risman theorems 1530. Introduction 153

CONTENTS 7

1. Preliminaries 1542. The proof 155

Chapter 9. Linked algebras 1570. Introduction 1571. The theorem 1582. Preliminary observations 1583. The proof 159

Chapter 10. Common splitting fields of division algebras 1630. Introduction 1631. “Generic” algebras of given behaviour 1642. Definition of β 1663. The theorem 166

Bibliography 169

8 CONTENTS

CHAPTER 1

Codimension 2 cycles on Severi-Brauer varieties

For a given sequence of integers (ni)∞i=1, we consider all the central simple

algebras A (over all fields) satisfying the condition indA⊗i = ni and findamong them an algebra having the biggest torsion in the second Chow groupCH2 of the corresponding Severi-Brauer variety (“biggest” means that it canbe mapped epimorphically onto each other).

We give a description of this biggest torsion in the general case (via thegamma-filtration) and find out when (i.e. for which sequences (ni)

∞i=1) it is

non-trivial. We also make an explicit computation in some special situations:e.g. in the situation of algebras of a square-free exponent e the biggest torsionturns out to be (cyclic) of order e.

As an application we prove indecomposability for certain algebras of aprime exponent.

0. Introduction

We consider finite dimensional central simple algebras over fields. Let Abe such an algebra and X = SB(A) the corresponding Severi-Brauer variety([6, §1]). We are interested to describe the torsion in the second Chow groupCH2(X) of 2-codimensional cycles onX modulo rational equivalence (the ques-tion seems more natural if one takes in account that the groups CH0(X) andCH1(X) never have a torsion). Here are some preliminary observations. Thegroup TorsCH2(X) is finite and annihilated by indA. Further, if A′ is anotheralgebra Brauer equivalent to A and X ′ = SB(A′) then by [58, Lemma 1.12] or[37, Corollary 1.3.2]

TorsCH2(X) ≃ TorsCH2(X ′) .

Finally, if

A =⊗p

Ap

is the decomposition of an algebra A into the tensor product of its primarycomponents and Xp = SB(Ap) for each prime p then

TorsCH2(X) ≃⊕p

TorsCH2(Xp)

or in other words, the p-primary part of the group TorsCH2(X) is isomorphicto TorsCH2(Xp) (Proposition 1.3).

9

10 1. CYCLES ON SEVERI-BRAUER VARIETIES

Summarizing, we see that the problem to compute TorsCH2(X) for allalgebras reduces itself to the case of primary division algebras.

Now consider the Grothendieck group K(X) = K0(X) together with thegamma-filtration (Definition 2.6):

K(X) = Γ0K(X) ⊃ Γ1K(X) ⊃ . . . .

One has a canonical epimorphism (see the proof of Corollary 2.15)

Γ2/3K(X) →→ CH2(X)

of the quotient

Γ2/3K(X) = Γ2K(X)/Γ3K(X) .

We consider the group Γ2/3K(X) as an upper bound for CH2(X) and will showthat in the primary case this upper bound is in certain sense the least one.

To formulate it precisely, let us call the sequence (indA⊗i)∞i=1 the behaviourof A. A sequence of integers (ni)

∞i=1 will be called a

((p-)primary

)behaviour if

it is the behaviour of a((p-)primary

)algebra.

Suppose that A is a division algebra. The Grothendieck group K(X) de-pends only on the behaviour of A (Theorem 3.1). Moreover, K(X) togetherwith the gamma-filtration (and the group Γ2/3K(X) in particular) depend onlyon the behaviour (Corollary 3.2) and our main observation is (Theorem 3.13):

For any primary behaviour (and any given field) there exists a division algebra

A (over an extension of the field) of the given behaviour for which the canonical

epimorphism Γ2/3K(X) →→ CH2(X) with X = SB(A) is bijective.

The construction of the algebra A is rather simple (Definition 3.12). Wetake a division algebra (over a suitable extension of the field) of the index as inthe given behaviour and of the exponent coinciding with the index. After thatwe pass to the function field of a product of certain generalized Severi-Brauervarieties in order to change the behaviour in the way prescribed.

Since the groups Γ2/3K(X) and CH2(X) have the same rank (Proposition2.14) (rank 1 if X is a Severi-Brauer variety of dimension at least 2), we alsohave an epimorphism of the torsion subgroups

Tors Γ2/3K(X) →→ TorsCH2(X)

which is moreover bijective iff Γ2/3K(X) →→ CH2(X) is (Corollary 2.15). So,formulating the main observation we may replace (and we do replace) boththe groups Γ2/3K(X) and CH2(X) by their torsion subgroups.

The gamma-filtration for a Severi-Brauer variety X and the group

Tors Γ2/3K(X)

in particular are from the so to say “algebro-geometrical” point of view veryeasy to compute (Propositions 4.1, 4.6, 4.10): K(X) is a subring of K(P)where P is a dimX-dimensional projective space and the Chern classes on X

1. REDUCTION TO THE PRIMARY CASE 11

with values in K (Definition 2.1, Remark 2.2) needed to determine the gamma-filtration are simply the restrictions of the Chern classes on K(P). Howeverto get the answer in a final form (say, to find the canonical decompositionof the finite abelian group Tors Γ2/3K(X) for any primary behaviour) furthercalculations are required which can be done e.g. by computer in every partic-ular situation (i.e. for every particular behaviour) but seem to be not easy inthe general case. Our main efforts in this direction are made in Propositions4.7, 4.9, 4.13 and 4.14 where we firstly find out when this group is non-trivial(Propositions 4.7, 4.9) and after that describe a wide class of situations whenthis group is cyclic and compute its order (Propositions 4.13, 4.14, see alsoExample 4.15).

To the structure of the chapter.In §1 we reduce the problem of computation of TorsCH2(SB(A)) for an

arbitrary central simple algebra A to the case when indA is a power of aprime. In §2 we recall and partially prove certain general facts on the Chernclasses (with various values) and on the gamma-filtration. In §3 we make themain observation. In §4 we investigate the group Γ2/3K for various primarybehaviours.

In §5 we consider algebras of prime exponent. We show that the groupTorsCH2(X), where X = SB(A) for an algebra A of a prime exponent p, is(cyclic) of order p or trivial (Proposition 5.1). Moreover, if A decomposes (intoa tensor product of two smaller algebras), then TorsCH2(X) = 0 (Proposition5.3). However, the torsion group is non-trivial if A is a “generic” division al-gebra of index pn and exponent p (see Example 4.12 for the definition) withn ≥ 2 for an odd p and n ≥ 3 for p = 2 (Proposition 5.1). Thus we obtaina wide family of indecomposable algebras (Corollary 5.4) which can be con-structed over an extension of any given field (without any restriction on thecharacteristic in particular). Here is a list of some articles where the questionof indecomposability for central simple algebras was considered previously:[4, 73, 83, 26, 35]. The method of [35] is close to but different from the onepresented here; it does not cover the case p = 2.

Some additional notations concerning filtrations on K(X) are introducedin § 2.

1. Reduction to the primary case

In this §, A is a central simple algebra over a field F , Ap (for every primenumber p) stays for the p-primary component of A, finally

X = SB(A) and Xp = SB(Ap) .

For an abelian group C, we denote by Cp its p-primary part.Let E/F be a finite field extension. Consider the homomorphisms

resE/F : CH2(X) → CH2(XE) and NE/F : CH

2(XE) → CH2(X) .


The projection formula shows that the composition NE/F ◦ resE/F coincideswith the multiplication by [E : F ].

Lemma 1.1. The composition resE/F ◦NE/F coincides with the multiplica-tion by [E : F ] as well.

Proof. Consider the homomorphisms resE/F and NE/F on the Grothen-dieck groups K(X) and K(XE). Since these groups are torsion-free and havethe same rank (Theorem 3.1) and since the composition NE/F ◦ resE/F is themultiplication by [E : F ], the composition taken in the other order is themultiplication by [E : F ] as well. Since the second Chow group coincides withthe second successive quotient of the topological filtration on the Grothendieckgroup (see e.g. [32, §3.1]), we are done.

Corollary 1.2. If [E : F ] is not divisible by a given prime number p,then

CH2(X)p ≃ CH2(XE)p .

Proposition 1.3. For every prime p, the p-primary part of the groupCH2(X) coincides with the torsion of CH2(Xp).

Proof. Fix a prime p and a finite field extension E/F of degree prime top such that the algebra AE is Brauer equivalent to (Ap)E. We have

CH2(X)p ≃ CH2(XE)p ≃ CH2((Xp)E)p ≃ CH2(Xp)p

(for the first and the third steps, we use the corollary). Since TorsCH2(Xp) isannihilated by indAp, we finally get

CH2(Xp)p = TorsCH2(Xp) .

2. Chern classes and gamma-filtration

In this §, we are working with the category of smooth projective irreduciblealgebraic varieties over a fixed field. The Grothendieck ring K is consideredas a contravariant functor on this category.

Definition 2.1 (Chern classes with values in K). The total Chern classct is a homomorphism of functors

ct : K+ −→ K[[t]]×

(where the left-hand side is the additive group of the ring K while the right-hand side is the multiplicative group of series in one variable t over K) satis-fying the following property: if ξ ∈ K(X) is a class of an invertible sheaf on avariety X then

ct(ξ) = 1 + (ξ − 1)t .

2. CHERN CLASSES AND GAMMA-FILTRATION 13

One defines the Chern classes ci : K → K by putting

ct =∞∑i=0

ci · ti .

Remark 2.2. Usually, one does not use the name “Chern classes” for themaps ci defined above e.g. since unlike the Chern classes 2.7, 2.8, and 2.11they do not satisfy the rule c1(ξ · η) = c1(ξ) + c1(η) for classes of invertiblesheaves ξ and η.

Proposition 2.3. Chern classes with values in K are unique.

Proof. Follows from the

Lemma 2.4 (Splitting principle, [52, Proposition 5.6]). For any varietyX and any x ∈ K(X) there exists a morphism f : Y → X such that:

1. f is a composition of some projective bundle morphisms;2. f ∗(x) ∈ K(Y ) is a linear combination (with integral coefficients) of

classes of some invertible sheaves.

To obtain uniqueness of the Chern classes just note that the homomorphismf ∗ : K(X) → K(Y ) in the lemma is injective.

Proposition 2.5. Chern classes with values in K exist.

Proof. Here is the way of constructing due to Grothendieck with theoriginal notations ([52, Theorem 3.10 and §8]).

Take a variety X. First one constructs a homomorphism

λt : K+ −→ K[[t]]×

by sending the class of a locally free sheaf E to

λt([E ]) =∞∑i=0

[ΛiE ] · ti

where ΛiE is the i-th exterior power of E .After that one considers another homomorphism

γt : K+ −→ K[[t]]×

namely,

γt = λ t1−t

(this γt gave the name of the gamma-filtration).Finally, one puts

ct = γt ◦ (id− rk)

where rk : K(X) → Z is the rank homomorphism (followed by the inclusionZ ↪→ K(X) more precisely).


Definition 2.6 (Gamma-filtration). The gamma-filtration

K(X) ⊃ Γ0K(X) ⊃ Γ1K(X) ⊃ . . .

is the smallest ring filtration on K(X) such that Γ0K(X) = K(X) and

ci(K(X)

)⊂ ΓiK(X) for all i ≥ 1 .

In other words, for every l ≥ 0, ΓlK(X) is the subgroup of K(X) generatedby all the products

ci1(x1) . . . cir(xr) with xj ∈ K(X) and

r∑i=1

ij ≥ l

(it might be not immediately clear but it is nevertheless easy to see that thegroup K(X) = Γ0K(X) is really also generated by these products).

In particular, Γ1K(X) = Ker(rk : K(X) → Z).We denote by G∗ΓK(X) the adjoint graded ring.

Definition 2.7 (Chern classes with values in G∗ΓK). For any varietyX,we call the induced maps

ci : K(X) → GiΓK(X)

the Chern classes with values in G∗ΓK. The total Chern class ct is the homo-morphism

ct : K(X)+ −→

(∞∑i=0

GiΓK(X) · ti)×

It is a morphism of functors and

ct(ξ) = 1 + (ξ − 1)t

for a class ξ ∈ K(X) of an invertible sheaf on X ((ξ − 1) is considered as anelement of G1ΓK(X) in the last formula).

Side by side with the gamma-filtration we consider the topological filtrationon K(X) (in fact defined on K ′

0(X)) ([69, §7]):K(X) = T0K(X) ⊃ T1K(X) ⊃ . . . .

Note that

T1K(X) = Ker(rk : K(X) → Z) = Γ1K(X) .

We will denote by G∗TK(X) the adjoint graded ring.

Definition 2.8 (Chern classes with values in G∗TK). The total Chernclass ct is a homomorphism of functors

ct : K+ −→

(∞∑i=0

GiTK · ti)×

satisfying the property:ct(ξ) = 1 + (ξ − 1)t .

2. CHERN CLASSES AND GAMMA-FILTRATION 15

One defines the Chern classes ci : K → GiTK by putting

ct =∞∑i=0

ci · ti .

Proposition 2.9. Chern classes with values in G∗TK are unique.

Proof. Follows from the splitting principle (Lemma 2.4) since the homo-morphism

f∗ : G∗TK(X) → G∗TK(Y )

is injective ([13, Lemma 3.8 of Chapter V]).

Proposition 2.10. Chern classes with values in G∗TK exist.

Proof. Simply compose the Chern classes with values in CH∗ (Definition2.11) with the canonical epimorphism CH∗ →→ G∗TK mapping a class [Z] ∈CH∗(X) of a simple cycle Z ⊂ X to the class of the structure sheaf OZ of Zprolonged to X by 0.

Definition 2.11 (Chern classes with values in CH∗). We repeat Defini-tion 2.8, replacing G∗TK by CH∗. In the formula ct(ξ) = 1 + (ξ − 1)t, weconsider (ξ − 1) as an element of CH1(X) via the canonical isomorphismCH1(X) ≃ G1TK(X) ([69, §7.5]) described in the proof of Proposition 2.10.

Proposition 2.12. Chern classes with values in CH∗ are unique.

Proof. Follows from the splitting principle (Lemma 2.4) since the homo-morphism

f ∗ : CH∗(X) → CH∗(Y )

is injective.

Proposition 2.13 ([12, §3.2]). Chern classes with values in CH∗ exist.

Now we establish certain connections between the gamma-filtration andthe topological one.

Proposition 2.14. For any variety X,

1. ΓiK(X) ⊂ TiK(X) for all i;2. ΓiK(X) = TiK(X) for i ≤ 2;3. ΓiK(X)⊗Q = TiK(X)⊗Q for all i.

Proof. 1. [13, Theorem 3.9 of Chapter V].2. We only need to manage the case i = 2.There are canonical isomorphisms

G1ΓK(X) ≃ Pic(X) ([13, Remark 1 in §3 of Chapter IV]) ;

CH1(X) ≃ G1TK(X) ([69, §7.5])(the definition of the second map is given in the proof of Proposition 2.10).

Since Γ2K(X) ⊂ T2K(X) we have a surjection G1ΓK(X) →→ G1TK(X)which gives an epimorphism Pic(X) → CH1(X). But the latter map is anisomorphism ([14, Corollary 6.16]). Thus Γ2K(X) = T2K(X).


3. [13, Proposition 5.5 of Chapter VI].

Corollary 2.15. One has an exact sequence

0 → T3K(X)/Γ3K(X) → TorsG2ΓK(X) → TorsCH2(X) → 0 .

Proof. The equality Γ2K(X) = T2K(X) and the inclusion Γ3K(X) ⊂T3K(X) stated in the proposition give an exact sequence

0 → T3K(X)/Γ3K(X) → G2ΓK(X) → G2TK(X) → 0 .

Consider the commutative diagram with exact columns

0 0↓ ↓

TorsG2ΓK(X)(1)−→ TorsG2TK(X)

↓ ↓G2ΓK(X)

(2)−→ G2TK(X)↓ ↓

G2ΓK(X)/Tors(3)−→ G2TK(X)/Tors

↓ ↓0 0

The map (2) is surjective. Hence the map (3) is surjective as well. Since bythe proposition the map (3)⊗Q is bijective, the map (3) itself is bijective aswell. Thus (1) is surjective and the kernels of (1) and (2) coincide. So, we getthe exact sequence

0 → T3K(X)/Γ3K(X) → TorsG2ΓK(X) → TorsG2TK(X) → 0 .

Finally, the canonical map CH2(X) → G2TK(X) is an isomorphism (see e.g.[32, §3.1], the definition of the map is given in the proof of Proposition 2.10).

As a corollary of the uniqueness assertion of Proposition 2.9, we get aconnection between Chern classes with different values:

Lemma 2.16. The following diagram of maps commutes:

K(X)ci−−−→ CHi(X)yci ycan.

GiΓK(X) −−−→ GiTK(X)

Proof. Both the compositions are Chern classes with values in G∗TK(Definition 2.8) which are unique (Proposition 2.9).

Remark 2.17. One can formulate a criterion (which will be used later) forwhen the gamma-filtration coincides with the topological one. It is clear fromthe very definition (Definition 2.6) that for any variety X the ring G∗ΓK(X)is generated by the Chern classes (with values in G∗ΓK). So, if the gamma

3. MAIN OBSERVATION 17

and the topological filtrations are the same, the ring G∗TK is generated bythe Chern classes (with values in G∗TK this time) as well.

The other way round, if G∗TK is generated by the Chern classes, then thehomomorphism G∗ΓK(X) → G∗TK(X) is surjective whence the filtrationscoincide.

3. Main observation

From now on, X denotes the Severi-Brauer variety corresponding to acentral simple algebra A over a field F .

Denote by P the projective space XF where F is an algebraic closure of Fand let ξ ∈ K(P) be the class of OP(−1). The ring K(P) is generated by ξsubject to only one relation:(ξ − 1)n = 0 where n = dimX + 1 = degA. Weconsider the restriction map K(X) → K(P) which is a ring homomorphismcommuting with the Chern classes 2.1.

Theorem 3.1 ([69, §8, Theorem 4.1]). The map K(X) → K(P) is injec-tive; its image is additively generated by (indA⊗i) · ξi (i ≥ 0).

Corollary 3.2. For a division algebra A, the group K(X) together withthe gamma-filtration depends only on the behaviour of A.

Proposition 3.3 ([33, Theorem 1]). If indA = expA for an algebra Athen (for any l ≥ 0) the l-th term of the topological filtration TlK(X) is gen-erated by all

indA

(i, indA)(ξ − 1)i with l ≤ i < degA

where (·, ·) denotes the greatest common divisor. In particular, the groupG∗TK(X) is torsion-free.

Proposition 3.4. If A is a primary algebra then

indA

(i, indA)(ξ − 1)i ∈ ΓiK(X) for any i ≥ 0 .

Proof. Put n = indA. For nξ ∈ K(X) we have:

ct(nξ) = ct(ξ)n =

(1 + (ξ − 1)t

)nwhere ct is the total Chern class with values in K (the last equality holds byDefinition 2.1). Whence

ci(nξ) =

(n

i

)(ξ − 1)i ∈ ΓiK(X) .

In particular,(ξ − 1)n ∈ ΓnK(X)

thereby for the rest of the proof we may assume that i ≤ n. Moreover,

ni(ξ − 1)i = c1(nξ)i ∈ ΓiK(X) .

The last observation is


Lemma 3.5. If n is a power of a prime p and n ≥ i ≥ 0 then(ni,

(n

i

))=

n

(i, n).

Moreover, if i = 0 then vp(ni

)= vp(n) − vp(i), where vp(i) is the multiplicity

of p in i.

Proof. The case i = 0 is evident. Suppose that i = 0. If 1 ≤ j < n thenvp(j) < vp(n) and so vp(n− j) = vp(j). Hence

vp

((n− 1

)(n− 2

). . .(n− (i− 1)

)1 · 2 . . . (i− 1)

)= 0

and

vp

(n

i

)= vp

(ni

)= vp

(n

(i, n)

).

Corollary 3.6. If A is a primary algebra and indA = expA then thegamma-filtration on K(X) coincides with the topological one.

Theorem 3.7. Let A be as in Corollary 3.6, X = SB(A). Let Y1, . . . , Ymbe some generalized Severi-Brauer varieties ([7, §4]) of some algebras whichare (Brauer equivalent to) some tensor powers of A.

The gamma-filtration on the Grothendieck group of the variety X over thefunction field F (Y1×· · ·×Ym) coincides with the topological one. In particular,the epimorphism of Corollary 2.15

TorsG2ΓK →→ TorsCH2

for this variety is bijective.

Proof. For every Yi the product X × Yi is a Grassman bundle over X(with respect to the first projection) (Corollary 6.4). Hence CH∗(X × Yi) isgenerated as a CH∗(X)-algebra by the Chern classes of a locally free sheaf (seee.g. [12, Proposition 14.6.5] or [43, Theorem 3.2]). Taking the product of allX × Yi over X we obtain that

CH∗(X × Y1 × · · · × Ym)

is generated as a CH∗(X)-algebra by the Chern classes (of some locally freesheaves).

The homomorphism of CH∗(X)-algebras

CH∗(X × Y1 × · · · × Ym) → CH∗(XF (Y1×···×Ym))

(given by the pull-back) is surjective (see e.g. [39, Theorem 3.1]). Whence theright-hand side is generated as a CH∗(X)-algebra by the Chern classes as well.

3. MAIN OBSERVATION 19

Using the epimorphism CH∗ →→ G∗TK of the Chow ring onto the adjointgraded Grothendieck ring we obtain the same statement as in the previousparagraph but for G∗TK instead of CH∗ by meaning the Chern classes withvalues in G∗TK this time.

The gamma-filtration on K(X) coincides with the topological one (Corol-lary 3.6) and therefore the ring G∗TK(X) is generated by the Chern classes(Remark 2.17). Consequently, G∗TK(XF (Y1×···×Ym)) is generated by the Chernclasses as a ring, not only as a G∗TK(X)-algebra. It means that the gamma-filtration onK(XF (Y1×···×Ym)) coincides with the topological one (Remark 2.17).

Definition 3.8. Let A be a p-primary algebra. The sequence of integers(logp indA

⊗pi)logp expAi=0

is called the reduced behaviour of A.

Example 3.9. The reduced behaviour of a p-primary algebra A with

indA = expA = pn

is n, n− 1, n− 2, . . . , 1, 0.

Proof. Suppose that n > 0. By [3, Lemma 7 on Page 76], indA⊗p <indA. Moreover, indA⊗p ≥ expA⊗p = pn−1. Thus indA⊗p = pn−1.

Lemma 3.10. The behaviour of a primary algebra is completely determinedby its reduced behaviour. The reduced behaviour of an algebra is a finite strongdecreasing sequence of integers with 0 in the end. Any finite strong decreasingsequence of integers with 0 in the end is for any prime p the reduced behaviourof a p-primary division algebra.

Proof. Let A be a p-primary algebra. If i is an integer prime to p thenthe splitting fields of the algebra A are the same as the splitting fields of thealgebra A⊗i. Therefore indA⊗i = indA, what proves the first sentence of thelemma.

If in addition indA = 1 then indA⊗p < indA ([3, Lemma 7 on Page 76]).It proves the second sentence.

Finally, fix a sequence n0 > n1 > · · · > nm = 0 and a prime p. Aconstruction of a p-primary algebra having the reduced behaviour (ni)

mi=0 is

given in [78, Construction 2.8]. This construction involves function fields ofusual Severi-Brauer varieties only. We describe another known constructionwhich involves function fields of generalized Severi-Brauer varieties as well andis more suitable for our purposes.

We start with a division algebra A (over a suitable field) for which

indA = expA = pn0 .

For each i = 1, 2, . . . ,m consider the generalized Severi-Brauer variety

Yi = SB(pni , A⊗pi)


(we define here SB(pni , A⊗pi) to be the variety of rank pni left ideals in A⊗pi ;

its function field is a generic extension making the index of A⊗pi to be equalto pni).

Finally, we denote the function field F (Y1×· · ·×Ym) by F and put A = AF .Using the index reduction formula [7, Theorem 5] or an improved version of

this formula [60, Formula I] one can easily show that the algebra A has thereduced behaviour (ni)

mi=0.

Remark 3.11. In the construction described in the proof above, it is notnecessary to use all of the varieties Yi: if ni = ni−1 − 1 for some i then thevariety Yi can be omitted.

Definition 3.12. We refer to an algebra A constructed like as in theabove proof (with taking the remark into account) as to a “generic” p-primarydivision algebra of the reduced behaviour (ni)

mi=0. Note that it can be con-

structed over an extension of any given field.

Theorem 3.13. Fix a prime p and a reduced behaviour. If A is a “generic”p-primary division algebra of the given reduced behaviour (Definition 3.12) thenthe epimorphism of Corollary 2.15

TorsG2ΓK(X) →→ TorsCH2(X) (where X = SB(A))

is bijective. If A is an arbitrary p-primary algebra of the same reduced be-

haviour as A then there exists an epimorphism

TorsCH2(X) →→ TorsCH2(X) .

Proof. The first part follows from Theorem 3.7 and from the definition of“generic” algebras (Definition 3.12). The second part follows from Corollary2.15, Corollary 3.2 and from the first one.

4. Computation of gamma-filtration

Remind that we have put forever X = SB(A).

Proposition 4.1. Let A be a p-primary algebra and (ni)mi=0 its reduced

behaviour. For any l ≥ 0, the group ΓlK(X) is generated by all the productsm∏i=0

pni

(ji, pni)(ξp

i − 1)ji with ji ≥ 0 andm∑i=0

ji ≥ l(∗)

where ξ = [O(−1)] ∈ K(P).

Proof. The formula

cj(pniξpi

) =

(pni

j

)(ξp

i − 1)j where 0 ≤ j ≤ pni

and Lemma 3.5 show that for any j ≥ 0 (even for j > pni) the elementpni

(j,pni )(ξp

i − 1)j lies in ΓjK(X) (compare with the proof of Proposition 3.4).

Therefore, each product (∗) lies in ΓlK(X).

4. COMPUTATION OF GAMMA-FILTRATION 21

For the opposite inclusion we need

Lemma 4.2. Consider the polynomials over Z in one variable ζ. For anyintegers j, r ≥ 0 the polynomial (ζr − 1)j is equal to a sum∑

s≥j

as(ζ − 1)s

with integers as such that s · as is a multiple of j · r.

Proof. It is clear that

(ζr − 1)j =

j·r∑s=j

as(ζ − 1)s

for some (uniquely determined) as ∈ Z. Taking the derivative we obtain thestatement on the coefficients.

As follows from Theorem 3.1, the additive group K(X) is generated by all

pniξrpiwith 0 ≤ i ≤ m and r ≥ 0. We have:

cj(pniξrpi

) =

(pni

j

)(ξrp

i − 1)j =∑s≥j

(pni

j

)· as · (ξp

i − 1)s .

Since by the lemma j | s · as and vp(pni

j

)equals ni − vp(j) or ∞ (Lemma 3.5),

the coefficient (pni

j

)· as is divisible by

pni

(s, pni).

Thus the Chern class cj(pniξrpi) is a linear combination (with integral coeffi-

cients) ofpni

(s, pni)(ξp

i − 1)s with s ≥ j .

Definition 4.3. Fix a prime number p, a reduced behaviour (ni)mi=0, and

consider a polynomial ring Z[ζ]. Let K ⊂ Z[ζ] be the additive subgroup

generated by all pni · ζr·pi where 0 ≤ i ≤ m and r ≥ 0. Consider a filtration Γon K defined by the formula of Proposition 4.1: for any l ≥ 0, the group ΓlKis generated by all the products

m∏i=0

pni

(ji, pni)(ζp

i − 1)ji with ji ≥ 0 andm∑i=0

ji ≥ l .(∗)

Note that K is a ring and that for any l1, l2 ≥ 0 one has Γl1K · Γl2K ⊂Γl1+l2K.

Lemma 4.4. In the notation of the definition, one has:

1. K = Γ0K ;2. for any l ≥ 0, K ⊃ ΓlK ⊃ Γl+1K ;3. if n is a multiple of pn0, then ΓnK = (ζ − 1)nK .


Proof. 1. The inclusion K ⊂ Γ0K is evident. The inverse inclusion is aparticular case of the second statement of the lemma.2. The inclusion ΓlK ⊃ Γl+1K is evident. Fix an arbitrary l ≥ 0. We prove theinclusion K ⊃ ΓlK using the third statement of the lemma. Choose a multiplen of pn0 such that n > l. Since ΓlK ⊃ ΓnK = (ζ − 1)nK ⊂ K , it suffices toshow that K/(ζ − 1)nK ⊃ ΓlK/(ζ − 1)nK . Find a p-primary algebra A (overan appropriate field F ) having the reduced behaviour (ni)

mi=0 (see Lemma 3.10

for the existence of A). The latter inclusion follows now from Proposition 4.1.3. Since (ζ − 1)n ∈ ΓnK, the inclusion ⊃ holds. One also sees immediatelyfrom the definition that any polynomial f ∈ ΓnK is of the kind f = (ζ−1)n ·hwhere h ∈ Z[ζ]. We have to prove that h ∈ K. It is a consequence of thefollowing

Lemma 4.5. Let f , g, and h are polynomials from Z[ζ] such that f = g ·hand suppose that the free coefficient of g equals ±1. If f and g lie in K then hlies in K as well.

Proof. Let

f =∑i≥0

fiζi , g =

∑i≥0

giζi , h =

∑i≥0

hiζi .

We prove that hiζi ∈ K using an induction on i. There is no problem

with the base of the induction: h0 ∈ K for any integral h0. Suppose thath1ζ, . . . , hi−1ζ

i−1 ∈ K. Polynomial f is equal to the product of g and h; be-cause of that we have:

fiζi = ±hiζ i + g1ζ · hi−1ζ

i−1 + · · ·+ gi−1ζi−1 · h1ζ + giζ

i · h0 .Since g ∈ K, every its monomial gjζ

j is in K as well (see the definition of K).By the same reason, fiζ

i ∈ K. Hence hiζi ∈ K.

If now A is a p-primary algebra of the reduced behaviour (ni)mi=0, the ring

homomorphism K → K(X) mapping ζ to ξ respects the filtrations and therebyinduces a homomorphism of graded groups

G∗ΓK → G∗ΓK(X) .

Proposition 4.6. For every 0 ≤ l < degA the group homomorphism

GlΓK → GlΓK(X)

is bijective.

Proof. It is evidently surjective by Proposition 4.1. To see the rest, putn = degA. By Lemma 4.4 and Theorem 3.1, the ring homomorphism ϕ :Γ0/nK → K(X) is bijective. Consider the induced filtration on Γ0/nK. Weknow that the bijective ring homomorphism ϕ respects the filtrations and issurjective on the successive quotients. Thus it is bijective on the successivequotients.


The proposition gives in particular a description of the group G2ΓK(X)for any p-primary algebra A of the reduced behaviour (ni)

mi=0. We want to find

out when this group has a non-trivial torsion. We start with the case of anodd prime.

Proposition 4.7. Let A be a p-primary algebra with an odd p. The groupG2ΓK(X) has a torsion iff indA > expA.

Proof. See Proposition 3.3 with Corollary 3.6 for the “only if” part.Suppose that indA > expA. Then in the reduced behaviour (ni)

mi=0 of A

one has:ns ≤ ns−1 − 2 for some s .

Using Proposition 4.6, we shall work with G2ΓK instead of G2ΓK(X).Consider the element

x = pns−1−2(ζps − 1)2 − pns−1(ζp

s−1 − 1)2 ∈ Γ2K .

Since x is divisible by (ζ − 1)3 in the polynomial ring Z[ζ], since moreoverpn0(ζ − 1)3 ∈ Γ3K and pn0f(ζ) ∈ K for any polynomial f(ζ) by Definition 4.3,one sees that a multiple of x lies in Γ3K. So, for our purposes it suffices toshow that x itself is not in Γ3K.

Let us act in the polynomial ring Z[ζ] modulo pns−1−1. We have:

x ≡ pns−1−2(ζps − 1)2 .

Consider a generator of Γ3K (Definition 4.3):m∏i=0

pni

(ji, pni)(ζp

i − 1)ji where ji ≥ 0 andm∑i=0

ji ≥ 3 .(∗)

We state that(∗) ≡ (ζp

s − 1)3 · f(ζps)where f is a polynomial. If we would manage to show it, we could proceed asfollows. Suppose that x ∈ Γ3K. Then

pns−1−2(ζps − 1)2 = (ζp

s − 1)3 · f(ζps) + pns−1−1 · g(ζps)for some polynomials f and g. Canceling by pns−1−2 and (ζp

s − 1)2 and sub-stituting t = ζp

s − 1 we get:

1 = tf0(t) + pg0(t) ∈ Z[t]what is a contradiction because t and p do not generate the unit ideal in thepolynomial ring Z[t].

It remains to show that

(∗) ≡ (ζps − 1)3 · f(ζps) .

If for all i < s the number ji in the product (∗) equals 0 then even the exactequality (not only the congruence) holds. Suppose that ji = 0 for some i < s.Write down this ji as ji = pr · j with j prime to p. If ni − r ≥ ns−1 − 1 then

pni

(ji, pni)≡ 0


and hence (∗) ≡ 0. So, assume that ni − r < ns−1 − 1. We have:

r > ni − ns−1 + 1 ≥ (s− 1)− i+ 1 = s− i .

In order to proceed we need

Lemma 4.8. In a polynomial ring Z[t], there is a congruence

(t− 1)pk ≡ (tp − 1)p

k−1

mod pk

for any prime p and any integer k > 0.

Proof. Induction on k starting from k = 1:

(t− 1)pk+1

=((t− 1)p

k)p

=

=((tp − 1)p

k−1

+ pk · f(t))p

≡ (tp − 1)pk

mod pk+1

(f(t) is a polynomial, it exists by the induction hypothesis).

According to the lemma we have:

(ζpi − 1)p

r ≡ (ζps − 1)p

r−s+imod pr−s+i+1 .

Hencepni

(ji, pni)(ζp

i − 1)ji ≡ pni

(ji, pni)(ζp

s − 1)pr−s+i·j mod pni−s+i+1 .

Since pr−s+i · j ≥ p ≥ 3 and ni − s+ i+ 1 ≥ ns−1 − 1 we are done.

The analogous statement in the case p = 2 looks out a little bit morecomplicated:

Proposition 4.9. Let A be a 2-primary algebra. The group G2ΓK(X)has a torsion iff indA > expA and the reduced behaviour of A is not of thekind

n, n− 1, . . . , 3, 2, 0 .

Proof. We start with the “only if” part. The case indA = expA iscovered by Proposition 3.3 with Corollary 3.6. Suppose that A has the reducedbehaviour

n, n− 1, . . . , 3, 2, 0 .

Using the same method as in [33] one can show that the whole adjoint gradedgroup is torsion-free. Namely, a formula like one of [33, Proposition] states:

|TorsG∗ΓK(X)| = |G∗ΓK(P)/ ImG∗ΓK(X)||K(P)/K(X)|

where | . | denotes the order of a group. Since we know the behaviour of A wecan compute that

|K(P)/K(X)| = 1

2

2n−1∏i=0

2n

(i, 2n)


(to avoid unnecessary complications we assume here that A is a division alge-bra). On the other hand, Proposition 3.4 shows that

|GiΓK(P)/ ImGiΓK(X)| ≤ 2n

(i, 2n)for any i .

Moreover,

|G1ΓK(P)/ ImG1ΓK(X)| ≤ 2n−1

because ξ2n−1 − 1 ∈ Γ1K(X) (see also the computation of CH1(X) in [6, §2])

and therefore

|G∗ΓK(P)/ ImG∗ΓK(X)| ≤ 1

2

2n−1∏i=0

2n

(i, 2n).

Thus, |TorsG∗ΓK(X)| = 1.Now we “correct” the “if” proof of the previous proposition in order to

match the current 2-primary situation. Suppose that we have an algebra A forwhich existence of the torsion is stated. Then in the reduced behaviour (ni)

mi=0

of A we have:

ns ≤ ns−1 − 2 and ns−1 ≥ 3 for some s .

Consider the element

x = 2ns−1−3(ζ2s − 1)2 − 2ns−1−1(ζ2

s−1 − 1)2 ∈ Γ2Kwhere K is as in Definition 4.3. Since in Z[ζ] the polynomial x is divisible by(ζ − 1)3, it is clear that a multiple of x lies in Γ3K. So, for our purposes itsuffices to show that x itself is not in Γ3K.

Let us act in the polynomial ring Z[ζ] modulo 2ns−1−2. We have:

x ≡ 2ns−1−3(ζ2s − 1)2 .

Consider a generator of Γ3K given in Proposition 4.1:m∏i=0

2ni

(ji, 2ni)(ζ2

i − 1)ji where ji ≥ 0 andm∑i=0

ji ≥ 3 .(∗)

We state that

(∗) ≡ (ζ2s − 1)3 · f(ζ2s)

where f is a polynomial. If we would manage to show it we could proceed inthe same manner as in the proof of the previous proposition.

If for all i < s the number ji in the product (∗) equals 0 then even theexact equality (not only the congruence) holds. Suppose that ji = 0 for somei < s. Write down this ji as ji = 2r · j with j prime to 2. If ni − r ≥ ns−1 − 2then

2ni

(ji, 2ni)≡ 0

and hence (∗) ≡ 0. So, assume that ni − r < ns−1 − 2. We have:

r > ni − ns−1 + 2 ≥ (s− 1)− i+ 2 = s− i+ 1 .


According to Lemma 4.8, we have:

(ζ2i − 1)2

r ≡ (ζ2s − 1)2

r−s+imod 2r−s+i+1 .

Hence2ni

(ji, 2ni)(ζ2

i − 1)ji ≡ 2ni

(ji, 2ni)(ζ2

s − 1)2r−s+i·j mod 2ni−s+i+1 .

Since 2r−s+i · j ≥ 22 ≥ 3 and ni − s+ i+ 1 ≥ ns−1 − 1 we are done.

Now we want to reduce the number of generators of the filtration of Defi-nition 4.3.

Proposition 4.10. In the notation of Definition 4.3, for every l ≥ 0, thegroup ΓlK is in fact also generated by a reduced number of the products (∗),namely by the products satisfying the additional condition: ji = 0 for every isuch that ni = ni−1 − 1.

Proof. Fix some i such that ni = ni−1 − 1. One has

(ζpi − 1)j =

∑s≥j

as · (ζpi−1 − 1)s

for some integers as with j · p | s · as (Lemma 4.2). Consequently,

pni

(j, pni)(ζp

i − 1)j =∑s≥j

as ·pni

(j, pni)(ζp

i−1 − 1)s =∑s≥j

bs ·pni−1

(s, pni−1)(ζp

i−1 − 1)s

for some integers bs.

Using the proposition, we compute the group TorsG2ΓK(X) explicitly ina special situation. The situation we mean is described in the following

Definition 4.11. We say that a reduced behaviour (ni)mi=0 “makes (ex-

actly) one jump” iff there exists exactly one s such that ns ≤ ns−1 − 2.

Example 4.12. Fix a prime p and integers n > m ≥ 1. One can define a

“generic” division algebra A of index pn and exponent pm in spirit of Definition3.12: take a division algebra A of index and exponent pn, put Y = SB(A⊗pm)

and A = AF (Y ).

The resulting algebra A can be also obtained as a “generic” p-primarydivision algebra of the reduced behaviour

n, n− 1, . . . , n−m+ 2, n−m+ 1, 0 .

In particular, it is an example of an algebra with reduced behaviour “makingone jump”.

Proposition 4.13. Let A be a p-primary algebra with an odd p and sup-pose that the reduced behaviour (ni)

mi=0 of A “makes one jump”. Then the

torsion in G2ΓK(X) is a cyclic group of order p to the power

min{s, n0 − ns − s}where s is the subscript for which ns ≤ ns−1 − 2.


Proof. We work with K instead ofK(X) (see Proposition 4.6). Accordingto Proposition 4.10, for any l ≥ 0, the group ΓlK is generated by the products:

pn0

(j0, pn0)(ζ − 1)j0 · pns

(js, pns)(ζp

s − 1)js with ji ≥ 0 and j0 + js ≥ l .

In particular, residue classes in the quotient G2ΓK of the following three ele-ments

u = pn0(ζ − 1)2 ;

v = pn0(ζ − 1) · pns(ζps − 1) ;

w = pns(ζps − 1)2

of Γ2K generate the quotient. The second one can be excluded: the differencev−pns+su is in Γ3K since it is divisible by pn0(ζ−1)3 in Z[ζ] and thereby can bewritten as a linear combination (with integral coefficients)of the polynomials

pn0(ζ − 1)3, pn0(ζ − 1)4, . . . ∈ Γ3K .

Since the classes of u and w in the quotient have infinite order, any torsionelement x ∈ G2ΓK of the kind x = u − kw or x = ku − w with an integer k(if exists) generates the torsion subgroup. Consider two cases: if n0 ≥ ns + 2sthen we put

x = u− pn0−ns−2sw ;

otherwise we put

x = pns+2s−n0u− w .

The element x ∈ G2ΓK is evidently a torsion element. We finish the proofwhen we show that x has order ps in the first case and order pn0−ns−s in thesecond. In both cases it means the same:

pn0+s(ζ − 1)2 − pn0−s(ζps − 1)2 ∈ Γ3K(1)

and

pn0+s−1(ζ − 1)2 − pn0−s−1(ζps − 1)2 ∈ Γ3K .(2)

In order to avoid repetition of some boring computations, we prove the in-clusion (1) in the following “tricky” way. Consider a ring K′ with a filtration Γconstructed as in Definition 4.3 for the reduced behaviour (n0, n0−1, . . . , 1, 0).The ring K′ is contained in K and this inclusion respects the filtrations. Theelement of (1) is in Γ2K′ and there is a multiple of it lying in Γ3K′. Since G∗ΓK′

is torsion-free (Propositions 4.6 and 3.3), this element lies even in Γ3K′. Hence(1).

The proof of (2) goes parallel to the proof of Proposition 4.7 and does notcontain any new idea. Let us act in the polynomial ring Z[ζ] modulo pn0−s.The element we are interested in is congruent to

pn0−s−1(ζps − 1)2 .


Consider a generator of Γ3K:

pn0

(j0, pn0)(ζ − 1)j0 · pns

(js, pns)(ζp

s − 1)js with ji ≥ 0 and j0 + js ≥ 3 .(∗)

The proof is completed when we show that

(∗) ≡ (ζps − 1)3 · f(ζps)

where f is a polynomial (compare with the proof of Proposition 4.7).If j0 = 0 then even the exact equality (not only the congruence) holds.

Suppose that j0 = 0. Write down j0 as j0 = pr · j with j prime to p. Ifn0 − r ≥ n0 − s then

pn0

(j0, pn0)≡ 0

and hence (∗) ≡ 0. So, assume that n0−r < n0−s, i.e. that r > s. Accordingto Lemma 4.8, we have:

(ζ − 1)pr ≡ (ζp

s − 1)pr−s

mod pr−s+1 .

Hence

pn0

(j0, pn0)(ζ − 1)j0 ≡ pn0

(j0, pn0)(ζp

s − 1)pr−s·j mod pn0−s+1 .

Since pr−s · j ≥ p ≥ 3 and n0 − s+ 1 ≥ n0 − s, we are done.

Proposition 4.14. Let A be a 2-primary algebra. Suppose that the re-duced behaviour (ni)

mi=0 of A “makes one jump” and let s be the subscript for

which ns ≤ ns−1 − 2. The group TorsG2ΓK(X) is cyclic; its order equals p tothe power {

min{s, n0 − ns − s} if ns > 0;

min{s, n0 − s− 1} if ns = 0.

Proof. We describe here only the changes which should be made in orderto adopt the previous proof to the 2-primary case.

First suppose that ns > 0.The quotient G2ΓK has three generators:

u = 2n0−1(ζ − 1)2 ;

v = 2n0(ζ − 1) · 2ns(ζ2s − 1) ;

w = 2ns−1(ζ2s − 1)2 .

The second one can be evidently excluded.If n0 ≥ ns + 2s then we put

x = u− 2n0−ns−2sw ;

otherwise we put

x = 2ns+2s−n0u− w .


The element x ∈ G2ΓK generates the torsion subgroup. To verify the state-ment on its order we have to check that

2n0+s−1(ζ − 1)2 − 2n0−s−1(ζ2s − 1)2 ∈ Γ3K(1)

and

2n0+s−2(ζ − 1)2 − 2n0−s−2(ζ2s − 1)2 ∈ Γ3K(2)

The inclusion (1) can be done in the same way as previously.Let us do (2). We act in the polynomial ring Z[ζ] modulo 2n0−s−1. The

element we are interested in is congruent to

2n0−s−2(ζ2s − 1)2 .

Consider a generator of Γ3K:

2n0

(j0, 2n0)(ζ − 1)j0 · 2ns

(js, 2ns)(ζ2

s − 1)js with ji ≥ 0 and j0 + js ≥ 3 .(∗)

The proof is complete when we show that

(∗) ≡ (ζ2s − 1)3 · f(ζ2s)

where f is a polynomial.If j0 = 0 then even the exact equality (not only the congruence) holds.

Suppose that j0 = 0. Write down j0 as j0 = 2r·j with odd j. If n0−r ≥ n0−s−1then

2n0

(j0, 2n0)≡ 0

and hence (∗) ≡ 0. So, assume that n0 − r < n0 − s − 1, i.e. that r > s + 1.According to Lemma 4.8, we have:

(ζ − 1)2r ≡ (ζ2

s − 1)2r−s

mod 2r−s+1 .

Hence2n0

(j0, 2n0)(ζ − 1)j0 ≡ 2n0

(j0, 2n0)(ζ2

s − 1)2r−s·j mod 2n0−s+1 .

Since 2r−s · j ≥ 22 ≥ 3 and n0 − s+ 1 ≥ n0 − s− 1 we are done.Now suppose that ns = 0.The generators of G2ΓK are:

u = 2n0−1(ζ − 1)2 ;

v = 2n0(ζ − 1) · (ζ2s − 1) ;

w = (ζ2s − 1)2 .

The second one can be evidently excluded.If n0 ≥ 2s+ 1 then we put

x = u− 2n0−2s−1w ;

otherwise we putx = 22s+1−n0u− w .


The element x ∈ G2ΓK generates the torsion subgroup. To verify the state-ment on its order we have to check that

2n0+s−1(ζ − 1)2 − 2n0−s−1(ζ2s − 1)2 ∈ Γ3K(1)

and

2n0+s−2(ζ − 1)2 − 2n0−s−2(ζ2s − 1)2 ∈ Γ3K(2)

But it was done already (the assumption ns > 0 was not in use).

Example 4.15. Let A be a “generic” division algebra of index pn and

exponent pm (Example 4.12). Put X = SB(A). From Theorem 3.13 and

Propositions 4.13 and 4.14, it follows that TorsCH2(X) is a cyclic group oforder p to the power{

min{m, n−m} for an odd p;

min{m, n−m− 1} for p = 2.

5. Algebras of prime exponent

Applying Theorem 3.13 and Propositions 4.13 and 4.14 to the case of aprime exponent we can state

Proposition 5.1. Let A be an algebra of a prime exponent p. Then thegroup TorsCH2(X) is trivial or (cyclic) of order p. It is trivial if indA = por indA | 4. It is not if A is a “generic” division algebra of index pn andexponent p (see Definition 3.12 or Example 4.12) where n ≥ 2 in the case ofodd p and n ≥ 3 in the case when p = 2.

Corollary 5.2. Let A be an algebra of a square-free exponent e. Thegroup TorsCH2(X) is (cyclic) of order dividing e; moreover, there exists an

algebra A of the exponent e with TorsCH2(X) of order e.

Proof. Follows from Propositions 1.3 and 5.1.

It would be interesting to list all algebras A of prime exponent with trivialTorsCH2(X). We can only describe a class of such algebras. In [34] it wasshown that any decomposable (into a tensor product of two smaller algebras)division algebra of index p2 and exponent p has no torsion in CH2(X) (infact, there is no torsion in the whole graded group G∗TK(X) ([34, Theorem1])). The 2-analogy of this fact was obtained in [36, Corollary 3.1]: anydecomposable division algebra of index 23 and exponent 2 has no torsion inCH2(X) (although non-trivial torsion may exist in G∗TK(X)). These factscan be generalized as follows:

Proposition 5.3. Let A be a division algebra of prime exponent. If Adecomposes then the group CH2(X) is torsion-free.

Proof. First consider the case when p = 2. We have a surjection

TorsG2ΓK(X) →→ TorsG2TK(X) ≃ TorsCH2(X) .

5. ALGEBRAS OF PRIME EXPONENT 31

The group from the left-hand side is cyclic (Proposition 4.13), its generator isrepresented by the element

x = pn(ξ − 1)2 − pn−2(ξp − 1)2 ∈ Γ2K(X) = T2K(X)

where pn = indA.Let A = A1 ⊗A2 be the decomposition of A into a product of two smaller

algebras. Assume that the base field F has no extensions of degree prime top (otherwise we can replace F by a maximal extension of prime to p degree;such a change has no effect on CH2(X), compare with Corollary 1.2). Take anextension E/F of degree [E : F ] = pn−2 such that ind(A1)E = ind(A2)E = p(one can obtain E/F by taking first an extension E1/F of degree [E1 : F ] =(indA1)/p for which ind(A1)E1 = p and extending E1 to E in such a way that[E : E1] = (indA2)/p and ind(A2)E = p). Consider an element

y = p2(ξ − 1)2 − (ξp − 1)2 ∈ T2K(XE) .

Since the algebra AE is Brauer equivalent to a decomposable division algebraof index p2 the group CH2(XE) is torsion-free ([34, Theorem 1]). Hence,y ∈ T3K(X). Taking the transfer of y we get:

NE/F (y) = pn(ξ − 1)2 − pn−2(ξp − 1)2 = x ∈ T3K(X) .

Consequently TorsCH2(X) = 0.Now consider the case p = 2.If indA = 4 then TorsCH2(X) = 0 (see e.g. Proposition 4.9 or use the

Albert theorem and [34, Theorem 1]). Suppose that indA ≥ 8.The group TorsG2ΓK(X) is cyclic (Proposition 4.14), its generator is rep-

resented by the element

x = 2n−1(ξ − 1)2 − 2n−3(ξ2 − 1)2 ∈ Γ2K(X) = T2K(X)

where 2n = indA.Let A = A1 ⊗A2 be the decomposition of A into a product of two smaller

algebras and indA1 ≥ indA2. Assume that the base field F has no extensionsof odd degree. Take an extension E/F of degree [E : F ] = 2n−3 such thatind(A1)E = 4 and ind(A2)E = 2 (one can obtain E/F by taking first anextension E1/F of degree [E1 : F ] = (indA1)/4 for which ind(A1)E1 = 4 andextending E1 to E in such a way that [E : E1] = (indA2)/2 and ind(A2)E = 2).Consider an element

y = 22(ξ − 1)2 − (ξ2 − 1)2 ∈ T2K(XE) .

Since the algebra AE is Brauer equivalent to a decomposable division algebraof index 23 the group CH2(XE) is torsion-free ([36, Corollary 3.1]). Hence,y ∈ T3K(X). Taking the transfer of y we get:

NE/F (y) = 2n−1(ξ − 1)2 − 2n−3(ξ2 − 1)2 = x ∈ T3K(X) .

Consequently TorsCH2(X) = 0.

Corollary 5.4. A “generic” algebra of prime exponent p and index pn

(Example 4.12) is always indecomposable excluding the Albert case: p = 2 = n.


Proof. Follows from Propositions 5.1 and 5.3.

6. Appendix

This § is included because we do not have an appropriate reference forCorollary 6.4. A particular case of Corollary 6.4 is proved in [64, Proposition4.7].

We start with certain preliminary observations concerning functors of pointsof algebraic varieties (schemes).

Let F be a field. Denote by F -alg the category of commutative associativeunital F -algebras. One refers to a covariant functor from F -alg to the categoryof sets as to an F -functor.

Let X be a scheme over F . For any R ∈ F -alg the set of R-points X(R)of X is by definition the set MorF (SpecR,X) of morphisms of schemes overF . This set is evidentely functorial in R, so we obtain an F -functor X calledthe functor of points of the scheme X. A morphism of F -schemes f : X → Ygives a natural transformation of their functors of points.

Proposition 6.1 ([63, Proposition 2 in §6 of Chapter 2]). Let X and Ybe F -schemes and let ϕ : X → Y be a natural transformation of their functorsof points. There exists a unique morphism of F -schemes f : X → Y inducingϕ.

Corollary 6.2. Two F -schemes X and Y are isomorphic iff there existsa natural transformation of the F -functors ϕ : X → Y such that for everyR ∈ F -alg the map of sets ϕ(R) : X(R) → Y (R) is bijective.

If additionally we are given morphisms X → Z and Y → Z to one more F -scheme Z, then the schemes X and Y are isomorphic over Z iff there exists anatural transformation ϕ as above commuting with the natural transformationsto the F -functor Z.

We need a couple more of natural definitions and trivial remarks.Let F be an F -functor supplied with a natural transformation F → G to

another F -functor G. For any R ∈ F -alg, the fibre of F over an R-point x of Gis by definition the inverse image of x with respect to the map F(R) → G(R);let us denote it by Fx.

Let F ′ be one more F -functor supplied with a natural transformation toG. Giving a natural transformation F → F ′ over G is equivalent to giving acollection of maps of sets Fx → F ′

x for every R ∈ F -alg and every x ∈ G(R)satisfying the evident functorial property: if R → S is a homomorphism inF -alg, x ∈ G(R), and if y ∈ G(S) is the image of x with respect to the mapG(R) → G(S), the following diagram commutes:

Fx −−−→ F ′xy y

Fy −−−→ F ′y

6. APPENDIX 33

Now everything is prepared to prove

Proposition 6.3. Let A be a central simple algebra over a field F , Xits Severi-Brauer variety, and Y a generalized Severi-Brauer variety SB(n,A)with some n ≥ 0 (see Remark 6.6).

The product X × Y considered over X (via the first projection) is isomor-phic (as a scheme over X) to the Grassman bundle IΓ(n,V) “of n-dimensonalsubspaces” of the canonical vector bundle V on X (see e.g. [80, Page 94] fora definition of the canonical vector bundle on a Severi-Brauer variety).

Proof. It suffices to show that for every R ∈ F -alg and every x ∈ X(R)there is a natural bijection of the sets (X × Y )x and IΓ(n,V)x. First of all wegive descriptions of the sets of R-points of the varieties under consideration(these descriptions are in fact the most natural definitions of the varieties, seee.g. [38]).

The set Y (R) consists of left ideals J of the R-algebra AR = A⊗F R havingthe following two properties:

1. the exact sequence of AR-modules

0 → J → AR → AR/J → 0

splits (in particular, J is a projective R-module);2. rk J = n where rk J is the R-rank of J devided by degA.

Analogously, the set X(R) consists of right ideals I of AR such that thesequence 0 → I → AR → AR/I → 0 splits and rk I = 1. For the rest of theproof, we fix R, an ideal I like that, and we set x = I ∈ X(R). Note thatAR = EndR I.

The fiber Vx of V over x is I; IΓ(n,V)x is the set of R-submodules N of Isuch that the sequence

0 → N → I → I/N → 0

splits and rkRN = n.Now it is clear that the Morita theory ([11, Theorem 4.29]) gives a canonical

bijection of the sets (X×Y )x = Y (R) and IΓ(n,V)x: N ∈ IΓ(n,V)x correspondsto the left ideal HomR(I,N) of (EndR I)

op = AR, where (EndR I)op is the

opposite algebra.

Corollary 6.4. In the condition of the proposition, put Ym = SB(n,A⊗m)for any m > 0. Then X × Ym is a Grassman bundle over X.

Proof. Two ways of proving are possible: one can adopt the proof of theproposition to this new setting, or one can argue as follows.

Put Xm = SB(A⊗m) and consider the morphism of varieties X → Xm givenby the following natural transformation of their functors of points: for everyR ∈ F -alg the map X(R) → Xm(R) puts an ideal I ∈ X(R) to its m-th tensor


(over R) power I⊗m ∈ Xm(R). In the cartesian square

X × Ym −−−→ Xm × Ymy yX −−−→ Xm

the right arrow is a Grassman bundle by the proposition. Hence the left arrowis a Grassman bundle as well.

Remark 6.5. It is possible to “spread out” the statement of the corollarya little bit replacing A⊗m by any Brauer equivalent central simple F -algebra.

Remark 6.6. In contrast to [7, §2] and in contrast to the definition ofSB(A), we define here SB(n,A) to be the variety of rank n left ideals in A.This variety is canonically isomorphic to the variety of rank degA − n rightideals in A [42, §1 of Chapter I]. Of course, it is also isomorphic to the varietyof rank n right ideals in the opposite algebra Aop.

CHAPTER 2

Codimension 2 cycles on products of Severi-Brauervarieties

We study the Chow group of 2-codimensional cycles on products of n Severi-Brauer varieties (n ≥ 2). We analyze more detailed

• the product of a biquaternion variety and a conic;• the product of two Severi-Brauer surfaces.

0. Introduction

In Chapter 1, we study the Chow group CH2 for one Severi-Brauer variety.Here, using the same methods, we study the same group for a direct productof Severi-Brauer varieties. The motivation for doing this work is given by thefollowing result of O. Izhboldin ([24]): the function field of a Severi-Brauervariety is universally excellent if and only if the index of the correspondingalgebra is not divisible by 4. To prove this result, he needed a certain infor-mation on the group CH2 of the product of a biquaternion variety and a conic(Theorem 6.1).

Certain results on the group CH2 of a product of Severi-Brauer varietieswas obtained in [66]. Its connection with the 3d Galois cohomology group([66, Theorem 4.1]) was established and an example of product of three conicswith torsion in CH2 was constructed ([66, Remark 6.1]).

The main and general result of this Chapter is Theorem 5.5 (with Corollary5.6). We apply it to products of two small-dimensional varieties (Theorems 6.1and 7.1); this way we obtain, in particular, new examples of torsion in CH2.

We use the following terminology and notation. By saying “A is an al-gebra”, we always mean that A is a central simple algebra over a field. Foran algebra A over a field F , we denote by [A] its class in the Brauer groupBr(F ) of F ; expA stays for the exponent, degA for the degree and indA forthe index of A.

The Severi-Brauer variety of an algebra A is denoted by SB(A). A varietyis always a smooth projective algebraic variety over a field; a sheaf over X isan OX-module. The Grothendieck ring of a variety X is denoted by K(X);

K(X) = Γ0K(X) ⊃ Γ1K(X) ⊃ . . . and K(X) = T0K(X) ⊃ T1K(X) ⊃ . . .

are respectively the gamma-filtration and the topological filtration on K(X);we use the notation G∗ΓK(X) and G∗TK(X) for the adjoint graded rings ofthese filtrations. There are certain relations between G∗ΓK(X), G∗TK(X),and the Chow ring CH∗(X) we use here; they can be found in §2 of Chapter 1.

35

36 2. CYCLES ON PRODUCTS OF SEVERI-BRAUER VARIETIES

1. Two preliminary results

Proposition 1.1. Let A1, . . . , An and B1, . . . , Bm be algebras over a fieldF such that the subgroups in Br(F ) generated by [A1], . . . , [An] and by [B1],. . . , [Bm] coincide. Then

TorsCH2(SB(A1)× · · · × SB(An)

)≃ TorsCH2

(SB(B1)× · · · × SB(Bm)

).

Proof. Set X = SB(A1)×· · ·×SB(An), Y1 = SB(B1). It suffices to showthat

TorsCH2(X) ≃ TorsCH2(X × Y1) .

Since X×Y1 → X is a projective space bundle (Proposition 5.3), one has ([14,§2 of Appendix A])

CH2(X × Y1) ≃ CH2(X)⊕ · · · ⊕ CH2−dimY1(X) .

The last observation is: for all i < 2, the group CHi(X) has no torsion (fori = 1 see [74, Lemme 6.3, (i)]).

Let p be a prime. For an algebra A as well as for an abelian group A, weare going to denote by A{p} the p-primary part of A.

Proposition 1.2. Let A1, . . . , An be algebras over a field. One has

CH2(SB(A1)×· · ·×SB(An)

){p} ≃ TorsCH2

(SB(A1{p})×· · ·×SB(An{p})

).

Proof. For n = 1, the assertion is proved in Proposition 1.3 of Chapter 1.The same proof works for n > 1.

2. Grothendieck group of product of Severi-Brauer varieties

Let A1, . . . , An be algebras over a field F , let X1, . . . , Xn be their Severi-Brauer varieties, and X = X1 × · · · × Xn. Fix a separable closure F of Fand put Xi = (Xi)F for each i. The varieties Xi are (isomorphic to) projectivespaces; denote by ξi the class inK(X) of the tautological sheaf of the projectivespace bundle

X →∏j =i

Xj .

The ring K(X) is generated by the elements ξ1, . . . , ξn subject to the relations

(ξ1 − 1)degA1 = · · · = (ξn − 1)degAn = 0 .

Consider the restriction K(X) → K(X) which is a ring homomorphism.

Theorem 2.1. The homomorphism K(X) → K(X) is injective; its imageis additively generated by the elements

ind(A⊗j11 ⊗ · · · ⊗ A⊗jn

n ) · ξj11 · · · ξjnnwith 0 ≤ j1 < degA1, . . . , 0 ≤ jn < degAn.

Proof. Use a generalized Peyre’s version [66, Proposition 3.1] of Quillen’scomputation of K-theory for Severi-Brauer schemes [69, Theorem 4.1 of §8] ntimes.

3. DISJOINT VARIETIES AND DISJOINT ALGEBRAS 37

Corollary 2.2. For algebras A1, . . . , An of fixed degrees, the ring K(X)with the gamma-filtration (and in particular the group TorsG2ΓK(X)) dependonly on the numbers ind(A⊗j1

1 ⊗ · · · ⊗ A⊗jnn ).

Proof. By the theorem, the numbers determine K(X) completely as asubring in K(X). The Chern classes with values in K (Definition 2.1 of Chap-ter 1) for X, which determine the gamma-filtration (Definition 2.6 of Chapter1), are the restrictions of the Chern classes for X.

3. Disjoint varieties and disjoint algebras

Definition 3.1. Let X1, . . . , Xn be arbitrary varieties over a field. Wesay that they are disjoint if the ring homomorphism

K(X1)⊗ · · · ⊗K(Xn) → K(X1 × · · · ×Xn) ,

induced by the pull-back homomorphisms

pr ∗i : K(Xi) → K(X1 × · · · ×Xn)

with respect to the projections pr i : X1 × · · · ×Xn → Xi, is an isomorphism.

Proposition 3.2. Let X1, . . . , Xn be disjoint varieties. The gamma-filt-ration on K(X1×· · ·×Xn) coincides with the filtration induced by the gamma-filtrations on K(X1), . . . , K(Xn).

Proof. Denote by X the product X1 × · · · × Xn and by Γ the inducedfiltration, where for each l ≥ 0, the term ΓlK(X) is going to be the subgroupof K(X) generated by the products

pr ∗1 Γl1K(X1) · · · pr ∗n ΓlnK(Xn)

for all l1, . . . , ln ≥ 0 with l1 + · · · + ln ≥ l. Since a pull-back homomorphismrespects the gamma-filtration, one has an inclusion ΓlK(X) ⊂ ΓlK(X). Letus prove the inverse inclusion. Since the gamma-filtration Γ on K(X) is thesmallest ring filtration having the properties Γ0K(X) = K(X) and cl(x) ∈ΓlK(X) for all x ∈ K(X) and l ≥ 1, where cl is the l-th Chern class withvalues in K (Definition 2.1 of Chapter 1), it suffices to show that

cl(x) ∈ ΓlK(X) .(∗)

Since the varieties X1, . . . , Xn are disjoint, the additive group of K(X) isgenerated by the products

x = pr ∗1(x1) · · · pr ∗n(xn)(∗∗)

where xi ∈ K(Xi) is the class of a locally free sheaf. Therefore it suffices tocheck the inclusion (∗) only for x of the form (∗∗). Since cl commutes withpr ∗i , one has

cl(pr ∗i (xi)

)∈ ΓlK(X) ,

and the last step of the proof is


Lemma 3.3. Let n,m, l ≥ 0. There exists a Z-polynomial fl((σi), (τj)

),

where σ1, . . . , σn and τ1, . . . , τm are variables, having two following properties:

• if x, y ∈ K(X) are classes of locally free sheaves over a variety X, theChern class cl(x · y) is equal to fl

(ci(x), cj(y)

);

• if one puts deg σi = i and deg τj = j, the degree of every monomial offl is at least l.

Proof. By the splitting principle ([52, Proposition 5.6]), it suffices toconsider the case where

x = ξ1 + · · ·+ ξn, y = η1 + · · ·+ ηm

with the classes of invertible sheaves ξi, ηj. For the total Chern class ct (Defi-nition 2.1 of Chapter 1), one has

ct(x) = ct(n∑i=1

ξi) =n∏i=1

(1 + (ξi − 1)t

)=

n∏i=1

(1 + ait) where ai = ξi − 1;

ct(y) = ct(m∑j=1

ηj) =m∏j=1

(1 + (ηj − 1)t

)=

m∏j=1

(1 + bjt) where bj = ηj − 1;

ct(xy) = ct(∑i,j

ξiηj) =∏i,j

(1 + (ξiηj − 1)t

)=∏i,j

(1 + (aibj + ai + bj)t

).

The class cl(xy) is (by definition) the coefficient of tl in ct(xy). This coefficientis evidently a polynomial in a1, . . . , an and b1, . . . , bm symmetric with respectto the variables (ai) and also symmetric with respect to the variables (bj)(notice that the degree of each monomial is at least l). Consequently, bythe main theorem on the symmetric polynomials, cl(xy) = fl

((σi), (τj)

)for a

polynomial fl, where (σi)ni=1 are the standard symmetric polynomials for (ai)

(σi is a homogeneous polynomial of degree i) and (τj)mj=1 are the standard

symmetric polynomials for (bj). The assertion of the lemma concerning thedegree is evidently satisfied. Finally, note that σi = ci(x) and τj = cj(y).

Corollary 3.4. Let X1, . . . , Xn be varieties with finitely generated Gro-thendieck groups (for instance, Severi-Brauer varieties). If the varieties aredisjoint and the groups G∗ΓK(X1), . . . , G

∗ΓK(Xn) are torsion-free, then thegroup G∗ΓK(X1 × · · · ×Xn) is torsion-free as well.

Proof. According to the proposition, the natural homomorphism

G∗ΓK(X1)⊗ · · · ⊗G∗ΓK(Xn) → G∗ΓK(X1 × · · · ×Xn)

is surjective. By our assumption, the group on the left-hand side is finitelygenerated and torsion-free; so, it is a free abelian group of finite rank. Thisrank coincides with the rank of the group on the right-hand side, because thevarieties are disjoint.

No we are going to understand what the condition of being disjoint meansfor Severi-Brauer varieties.

4. “GENERIC” VARIETIES 39

Definition 3.5. Let A1, . . . , An be algebras over a field. We say that theyare disjoint if

ind(A⊗j11 ⊗ · · · ⊗ A⊗jn

n ) = indA⊗j11 · · · indA⊗jn

n for all j1, . . . , jn ≥ 0.

Proposition 3.6. Algebras A1, . . . , An are disjoint if and only if theirSeveri-Brauer varieties are disjoint.

Proof. Since, for an arbitrary algebra A, there is a canonical isomorphismK(A) = indA ·Z, where K(A) denotes the Grothendieck group of the algebra,the algebras are disjoint if and only if the maps

K(A⊗j11 )⊗ · · · ⊗K(A⊗jn

n ) → K(A⊗j11 ⊗ · · · ⊗ A⊗jn

n )

are isomorphisms for all 0 ≤ j1 < degA1, . . . , 0 ≤ jn < degAn. Taking thedirect sum over all j1, . . . , jn, we obtain the map(

degA1−1⨿j1=0

K(A⊗j11 )

)⊗ · · · ⊗

(degAn−1⨿jn=0

K(A⊗jnn )

)−→

−→degA1−1⨿j1=0

. . .degAn−1⨿jn=0

K(A⊗j11 ⊗ · · · ⊗ A⊗jn

n ) .

Identifying the factors of the product on the left-hand side with

K(SB(A1)

), . . . , K

(SB(An)

)and the direct sum on the right-hand side with

K(SB(A1)× · · · × SB(An)

)by Theorem 2.1, one obtains on the place of the arrow the homomorphism ofDefinition 3.1.

4. “Generic” varieties

Definition 4.1. Let us say that a variety X is “generic”, if the gamma-filtration on K(X) coincides with the topological filtration.

Lemma 4.2. If TorsG∗ΓK(X) = 0 (for an arbitrary variety X), then Xis “generic”.

Proof. To see that the filtrations coincide, it suffices to show that thehomomorphism

α : G∗ΓK(X) → G∗TK(X) ,

induced by the inclusion of the filtrations, is injective. Since α⊗Q is bijective([13, Proposition 5.5 of Chapter VI]), the kernel of α contains only elementsof finite order. Therefore, α is really injective if the group G∗ΓK(X) has notorsion.

Lemma 4.3. Let G → X be a grassmanian bundle. If X is “generic”, thevariety G is “generic” as well.


Proof. Since G is a grassmanian bundle over X, the CH∗(X)-algebraCH∗(G) is generated by the Chern classes (with values in CH∗) (see [12,Proposition 14.6.5] or [43, Theorem 3.2]). Using the natural epimorphismCH∗ → G∗TK, one obtains the same result for G∗TK: the G∗TK(X)-algebraG∗TK(G) is generated by the Chern classes (with values in G∗TK). Since X is“generic”, the ring G∗TK(X) itself is generated by the Chern classes (Remark2.17 of Chapter 1). Consequently, G∗TK(G) is generated by the Chern classesnot only as algebra but also as a ring. That means G is “generic” (Remark2.17 of Chapter 1).

Lemma 4.4. Let X → Y be a smooth morphism of varieties and let Xbe its generic fiber. If X is “generic”, the variety X (it is a variety over thefunction field of Y ) is also “generic”.

Proof. The morphism (of schemes) X → X induces a homomorphismof Grothendieck groups K(X) → K(X), respecting the both filtrations, anda homomorphism of Chow groups CH∗(X) → CH∗(X) which is surjective(Proposition 4.1 of Chapter 5, see also [39, Theorem 3.1]). Consequently, thehomomorphism

G∗TK(X) → G∗TK(X)

is also surjective, and therefore, for every l, the group TlK(X) is mappedsurjectively onto TlK(X). Since TlK(X) = ΓlK(X), it follows that TlK(X) ⊂ΓlK(X). The inverse inclusion is always true.

Corollary 4.5. Let X and Y be varieties over a field F such that theprojection X×Y → X is a grassmanian bundle. If X is “generic”, then XF (Y )

is also “generic”.

Proof. The variety X×Y is “generic” according to Lemma 4.3; thereforethe variety XF (Y ) is “generic” by Lemma 4.4.

5. “Generic” algebras

Proposition 5.1. Let A be a primary algebra (i.e. degA is a power of aprime). Suppose that

• either indA = expA• or indA = 2n and indA⊗2n−2

= 4 (n ≥ 2)

(an example of such A is a biquaternion algebra). Then the group G∗Γ(SB(A)

)is torsion-free.

Proof. For algebras of the first type see Proposition 3.3 and Corollary 3.6of Chapter 1; for the second type see the proof of Proposition 4.9 of Chapter 1.

Corollary 5.2. Let A1, . . . , An be disjoint algebras and suppose that eachAi satisfies the condition of Proposition 5.1. Then for the product X oftheir Severi-Brauer varieties, one has: TorsG∗ΓK(X) = 0; in particular,TorsCH2(X) = 0.

5. “GENERIC” ALGEBRAS 41

Proof. It is a straightforward consequence of the proposition with Corol-lary 3.4 and Proposition 3.6.

For an algebra B and an integer r ≥ 0, denote by SB(r, B) the general-ized Severi-Brauer variety of rank r right ideals in B ([7, §2]). In particular,SB(1, B) = SB(B).

Proposition 5.3. Let A1, . . . , An and B be algebras over a field, let X =SB(A1)× · · · × SB(An) and let Y = SB(r, B) with certain r ≥ 0.

If the Brauer class [B] of the algebra B belongs to the group generated by[A1], . . . , [An], then the projection X × Y → X is an r-grassmanian.

Proof. We may assume that

B ≃ A⊗j11 ⊗ · · · ⊗ A⊗jn

n

with some j1, . . . , jn ≥ 0. Consider the cartesian square

X × Y −−−→ T × Yy yX −−−→ T

where T = SB(B) and where the morphism X → T is given by tensor productof ideals. The arrow on the right-hand side (that is the projection T ×Y → T )is an r-grassmanian by Proposition 6.3 of Chapter 1. Therefore, the projectionX ×Y → Y (that is the left-hand side arrow) is an r-grassmanian as well.

Definition 5.4. We call a collection of algebras A1, . . . , An “generic”,if it can be obtained by the following procedure. One starts with disjointalgebras A1, . . . , An over a field F such that each Ai satisfies the condition ofProposition 5.1. Then one takes F -algebras B1, . . . , Bm such that their classesin Br(F ) belong to the subgroup generated by [A1], . . . , [An]. Finally, one takesas Y a direct product of some generalized Severi-Brauer of algebras B1, . . . , Bm

and one puts Ai = (Ai)F (Y ) for all i = 1, . . . , n.

Theorem 5.5. If a collection of algebras A1, . . . , An is “generic”, then theproduct X of their Severi-Brauer varieties is a “generic” variety (Definition4.1); in particular, the epimorphism

TorsG2ΓK(X) → TorsCH2(X)

is bijective in this case.

Proof. Let A1, . . . , An be algebras used in construction of our “generic”collection (Definition 5.4). Put Xi = SB(Ai) for i = 1, . . . , n and let X =X1×· · ·×Xn. According to Corollary 5.2, the group G∗ΓK(X) is torsion-free.In particular, the variety X is “generic” (Lemma 4.2).

Now, let Y be the direct product of generalized Severi-Brauer varieties, usedin the construction of our generic collection. By Proposition 5.3, the projectionX × Y → X is a fiber product (over X) of grassmanians. Therefore, usingCorollary 4.5 m times, one proves that the variety X = XF (Y ) is “generic”.


Corollary 5.6. Let A1, . . . , An be arbitrary algebras and let X be theproduct of their Severi-Brauer varieties. Let A1, . . . , An be a “generic” collec-tion of algebras such that deg Ai = degAi and

ind(A⊗j11 ⊗ · · · ⊗ A⊗jn

n ) = ind(A⊗j11 ⊗ · · · ⊗ A⊗jn

n )

for all i and all j1, . . . , jn. Then the group TorsCH2(X) is isomorphic to afactorgroup of TorsCH2(X).

Proof. By the theorem, there is an isomorphism

TorsCH2(X) ≃ TorsG2ΓK(X) ;

by Corollary 2.2, one has

TorsG2ΓK(X) ≃ TorsG2ΓK(X) ;

finally, we always have a surjection (Corollary 2.15 of Chapter 1)

TorsG2ΓK(X) →→ TorsCH2(X) .

6. Biquaternion variety times conic

A Severi-Brauer variety of a biquaternion algebra is called biquaternionvariety here.

Theorem 6.1. Let X be a biquaternion variety, Y be a conic (over thesame field) and A,B be the corresponding algebras (B is a quaternion algebra).

1. The torsion in the group CH2(X × Y ) is either trivial, or of order 2.2. If the torsion is non-trivial, then

indA = ind(A⊗B) = 4 and indB = 2 .(∗)

3. If the collection A,B is “generic” (Definition 5.4) and satisfies the con-dition (∗), then the torsion is not trivial.

Proof. If indB = 2, i.e. if B is split, then we know from Proposition 1.1that TorsCH2(X ×Y ) ≃ CH2(X); hereby the latter group is torsion-free ([34,Corollary]).

If indA = 4, than A is Brauer-equivalent to a quaternion algebra A′;denoting by X ′ its Severi-Brauer variety, one gets (Proposition 1.1)

TorsCH2(X × Y ) ≃ TorsCH2(X ′ × Y ) .

Since dim(X ′ × Y ) = 2, the group

G2ΓK(X ′ × Y ) = Γ2K(X ′ × Y ) ⊂ K(X ′ × Y )

has no torsion. It follows that in this case TorsCH2(X × Y ) = 0 as well.Let C be a division algebra Brauer-equivalent to the product A⊗ B; T =

SB(C). Using Proposition 1.1 once again, we have

TorsCH2(X × Y ) ≃ TorsCH2(T × Y ) .

6. BIQUATERNION VARIETY TIMES CONIC 43

If ind(A ⊗ B) ≤ 2, then dimT × Y ≤ 2 and we are done in the same way asabove.

If ind(A ⊗ B) = 8, then the algebras A,B are disjoint and Corollary 5.2shows that TorsCH2(X × Y ) = 0.

The rest is served by

Proposition 6.2. Suppose that a biquaternion algebra A and a quater-nion algebra B are division algebras and ind(A⊗ B) = 4. For X,Y as above,one has: TorsG2ΓK(X × Y ) ≃ Z/2.

Proof. Put K = K(X×Y ), K = K(X× Y ). The commutative ring K isgenerated by elements ξ, η subject to the relations (ξ− 1)4 = 0 = (η− 1)2 (see§2). In particular, the additive group of K is a free abelian group generatedby the elements ξiηj, i = 0, 1, 2, 3, j = 0, 1. We are also going to use anothersystem of generators: f igj, i = 0, 1, 2, 3, j = 0, 1, where f = ξ − 1, g = η − 1.

For each l, the l-th term ΓlK of the gamma-filtration on K is generatedby the products f igj with i + j ≥ l. In particular, GlΓK is an abelian groupfreely generated by the residue classes of the products f igj with i+ j = l.

Lemma 6.3. The subring K ⊂ K is additively generated by the elements

1, 4ξ, ξ2, 4ξ3, 2η, 4ξη, 2ξ2η, 4ξ3η .

Proof. It is a particular case of Theorem 2.1.

Lemma 6.4. The following elements are also generators of the additivegroup of K:

1, 2f − f2, 2g, 2f2, 4fg, 4f 3, 2f2g , 4f3g

(the singled out element is going to produce the torsion — see Corollary 6.9).

Proof. A straightforward verification.

Lemma 6.5. There are the following inclusions:

Γ1K ∋ 2f − f2, 2g ;

Γ2K ∋ 2f 2, 4fg, 2f 2g ;

Γ3K ∋ 4f 3, 2 · 2f 2g ;

Γ4K ∋ 4f 3g .

Proof. The assertion on Γ1K is evident.Since 2f 2, 4fg ∈ K ∩ Γ2K and the restriction homomorphism G1ΓK →

G1ΓK is injective ([74, Lemme 6.3, (i)]), the assertion on Γ2K holds (a directverification (see the rest of the proof) is also easy).

Finally, one has:

ct(4ξ) = (1 + ft)4 ⇒ c3(4ξ) = 4f3 ⇒ 4f 3 ∈ Γ3K ;

2f2 ∈ Γ2K and 2g ∈ Γ1K ⇒ 4f 2g = (2f2) · (2g) ∈ Γ3K ;

c4(4ξη) = (ξη − 1)4 =((f + 1)(g + 1)− 1

)4= 4f3g ∈ Γ4K .


Corollary 6.6. Denote by α∗ the restriction homomorphism

G∗ΓK → G∗ΓK .

For all i > 0, one has: Imαi ⊂ 2GiΓK.

Proof. According to the lemma, the group G1ΓK is generated by theresidue classes of the elements 2f−f 2 and 2g; their images in G1ΓK are reallydivisible by 2. So,the assertion of the corollary for i = 1 is proved.

Since the elements of Γ2K, Γ3K and Γ4K, listed in the lemma, generateΓ2K and are divisible by 2 in K, we obtain the assertion for i ≥ 2 (use theabsence of torsion in G∗ΓK).

Corollary 6.7. #(TorsG∗ΓK) ≤ 2.

Proof. Since the group G∗ΓK is torsion-free, TorsG∗ΓK ⊂ Kerα∗. Weare going to show that #(Kerα∗) ≤ 2, using the following formula ([33, Propo-sition]):

#(Kerα∗) = #(Cokerα∗)/#(K/K) .

It is easy to calculate that #(K/K) = 210. According to the lemma,

#(Cokerα∗) ≤ 211 .

Lemma 6.8. 2f2g ∈ Γ3K.

Proof. It suffices to show that Imα3 ⊂ 4G3ΓK.The group Imα3 is generated by the subgroup Imα1 · Imα2 and by the

subset α3(c3K), where c3 is the 3d Chern class with values in G∗ΓK (Def-inition 2.7 of Chapter 1). Since Imαi ⊂ 2GiΓK for i > 0 by Corollary6.6, one has: Imα1 · Imα2 ⊂ 4G3ΓK. Therefore, it suffices to verify thatα3(c3(S)

)⊂ 4G3ΓK for a system S of generators of the additive group of K.

The verification is trivial if we take as S the system of generators of Lemma6.3.

Corollary 6.9. The residue of 2f2g in G2ΓK has order 2 and generatesthe torsion subgroup.

Proof. The residue is of order 2 by Lemmas 6.8 and 6.5. It generates thewhole torsion subgroup (not only in G3ΓK but also in G∗ΓK) by Corollary6.7.

The proofs of the theorem and of the proposition are complete.

Remark 6.10. In the condition of the theorem, denote the base field by Fand suppose that there exists a quadratic extension L/F (or, more generally,an extension of degree not divisible by 4) such that the algebra AL is no more adivision algebra and the algebra BL is split. In this case, f 2g ∈ T3K(XL×YL);

7. PRODUCT OF TWO SEVERI-BRAUER SURFACES 45

using the norm map, we obtain: 2f2g ∈ T3K(X×Y ), i.e. TorsCH2(X×Y ) =0.

Therefore, if A,B are such that TorsCH2(X×Y ) = 0 (for example, if A,Bform a “generic” collection (Theorem 6.1)), there are no extensions like that.The first example of this phenomenon is constructed in [51].

7. Product of two Severi-Brauer surfaces

A Severi-Brauer surface is a Severi-Brauer variety of dimension 2.

Theorem 7.1. Let X,Y be Severi-Brauer surfaces over a field and letA,B be the corresponding algebras.

1. The torsion in the group CH2(X × Y ) is either trivial, or of order 3.2. If the torsion is not trivial, then

indA = indB = ind(A⊗B) = ind(A⊗Bo) = 3(∗)

where Bo is the algebra opposite to B.3. If the collection A,B is “generic” (Definition 5.4) and satisfies the con-

dition (∗), then the torsion is not trivial.

Proof. If at least one of the algebras A, B, A⊗ B, A⊗ Bo is split, thenthere exists an algebra C of degree 3 such that its class [C] in the Brauergroup generates the same subgroup as [A] and [B] (together). According toProposition 1.1, in this case, the group TorsCH2(X × Y ) is isomorphic to thegroup TorsCH2

(SB(C)

)which is trivial by [33, Corollary], or also by Lemma

2.4 of Chapter 4.If ind(A⊗ B) = ind(A⊗ Bo) = 9, then the algebras A,B are disjoint and

one can use Corollary 5.2.Put Y o = SB(Bo). Since by Proposition 1.1

TorsCH2(X × Y ) ≃ TorsCH2(X × Y o) ,

it suffices to consider only one of the two following cases:

• ind(A⊗B) = 3 and ind(A⊗Bo) = 9;• ind(A⊗B) = 9 and ind(A⊗Bo) = 3.

Lemma 7.2. If indA = indB = ind(A ⊗ B) = 3 and ind(A ⊗ Bo) = 9,then TorsG2ΓK(X × Y ) = 0.

Proof. Put K = K(X×Y ), K = K(X× Y ). The commutative ring K isgenerated by elements ξ, η subject to the relations (ξ− 1)3 = 0 = (η− 1)3 (see§2). In particular, the additive group of K is an abelian group freely generatedby the elements ξiηj , i, j = 0, 1, 2. We also are going to use another systemof generators: f igj, i, j = 0, 1, 2, where f = ξ − 1, g = η − 1.

For every l, the l-th term ΓlK of the gamma-filtration on K is generatedby the products f igj with i+ j ≥ l.


The condition of the lemma implies that

indA⊗2 = indB⊗2 = ind(A⊗2 ⊗B⊗2) = 3 and

ind(A⊗B⊗2) = ind(A⊗2 ⊗B) = 9 .

So, according to Theorem 2.1, the subring K ⊂ K is additively generated by

1, 3ξ, 3ξ2, 3η, 3ξη, 9ξ2η, 3η2, 9ξη2, 3ξ2η2 .

We are also going to use another system of generators:

1, 3f, 3g, 3f2, 3fg, 3g2, 9f2g, 3f2g + 3fg2 + 6f2g2, 9f 2g2 .

Now it is evident that the intersection K ∩ Γ3K is generated by

9f 2g, 3f 2g + 3fg2 + 6f 2g2, and 9f 2g2 .

To prove that the group G2ΓK is torsion-free, it suffices to verify that thesethree elements belong to Γ3K.

Since 3f 2, 3g2 ∈ Γ2K, and 3g ∈ Γ1K, one has:

9f 2g = (3f 2) · (3g) ∈ Γ3K, 9f 2g2 = (3f2) · (3g2) ∈ Γ4K .

The last element coincides with a 3-d Chern class:

c3(3ξη) = (ξη − 1)3 =((f + 1)(g + 1)− 1

)3= (fg + f + g)3 =

3fg(f + g)2 + (f + g)3 = 6f 2g2 + 3f 2g + 3fg2 .

We finish the proof of the theorem by

Proposition 7.3. If indA = indB = ind(A ⊗ B) = ind(A ⊗ Bo) = 3,then TorsG2ΓK(X × Y ) ≃ Z/3.

Proof. We use the notation introduced in the beginning of the proof ofthe last lemma.

Lemma 7.4. The subring K ⊂ K is now generated by 1 and 3K. More-over,

Γ1K = 3Γ1K ;

Γ2K = 3Γ2K ;

Γ3K ∋ 3f 2g − 3fg2, 3f2g + 3fg2 + 6f2g2 ;

Γ4K ∋ 9f 2g2 .

Proof. The assertion about Γ1K is trivial. The assertion about Γ2K iscaused by injectivity of the restriction homomorphism G1ΓK → G1ΓK ([74,Lemme 6.3, (i)]); 9f 2g2 ∈ Γ4K because 3f 2, 3g2 ∈ Γ2K.

To prove the assertion about Γ3K, let us compute the 3d Chern class

c3(ξ2η) = (ξ2η − 1)3 =((f + 1)2(g + 1)− 1

)3= 27f2g2 + 12f2g + 6fg2 .

Since 9f 2g, 9fg2 ∈ Γ3K, we conclude that 3f 2g − 3fg2 ∈ Γ3K.

7. PRODUCT OF TWO SEVERI-BRAUER SURFACES 47

Finally, as we have already computed in the proof of Lemma 7.2,

3f 2g + 3fg2 + 6f2g2 = c3(3ξη) ∈ Γ3K .

Corollary 7.5. #(TorsG∗ΓK) ≤ 3.

Proof. Analogously to Corollary 6.7. Now one has (Lemma 7.4):

#(K/K) = 38 and #(Cokerα∗) ≤ 39 .

Lemma 7.6. 3f2g2 ∈ Γ3K.

Proof. Let us define a homomorphism ϕ9 : K → Z/9 as follows: write anarbitrary element x ∈ K as a linear combination

x =2∑

i,j=0

aijfigj with aij ∈ Z,

put ϕ(x) = a21 + a12 − a22 and define ϕ9(x) as the residue of ϕ(x) modulo 9.Since ϕ9(3f

2g2) = 0, it suffices to show that ϕ9(Γ3K) = 0.

A priori, the group Γ3K is generated by Γ1K · Γ2K, c3(S) et c4(S) where

S = 1, 3ξ, 3ξ2, 3η, 3ξη, 3ξ2η, 3η2, 3ξη2, 3ξ2η2 .

Hereby, c4(s) = 0 for all s ∈ S; thus one can eliminate c4(S) from the list ofgenerators.

SinceΓ1K · Γ2K ⊂ Γ1K · Γ1K ⊂ 9K (Lemma 7.4),

one has: ϕ9(Γ1K · Γ2K) = 0.

It remains c3(S). For s = 1, 3ξ, 3ξ2, 3η, and 3η2, the value ϕ(s) is already 0.The following calculations show that ϕ9

(c3(s)

)= 0 for the other four elements

s ∈ S as well:

c3(3ξη) = (ξη − 1)3 =((f + 1)(g + 1)− 1

)3=(fg + (f + g)

)3=

= 3fg(f + g)2 + (f + g)3 = 6f2g2 + 3f 2g + 3fg2 ;

c3(3ξ2η2) = (ξ2η2−1)3 =((f+1)2(g+1)2−1

)3=((f 2+4fg+g2)+2(f+g)

)3=

= 12(f2 + 4fg + g2)(f + g)2 + 8(f + g)3 = 120f 2g2 + 24f2g + 24fg2 ;

c3(3ξ2η) = (ξ2η − 1)3 =((f + 1)2(g + 1)− 1

)3=(f(f + 2g) + (2f + g)

)3=

= 3f(f + 2g)(2f + g)2 + (2f + g)3 = 27f 2g2 + 12f 2g + 6fg2 ;

c3(3ξη2) = 27f 2g2 + 6f 2g + 12fg2 .


According to Lemma 7.4, we have 3f 2g2 ∈ Γ2K. The residue class of theelement 3f 2g2 in G2ΓK has order 3 by Lemmas 7.4 and 7.6. Therefore, byCorollary 7.5, it generates the whole torsion subgroup of G2ΓK. So, the proofof Proposition 7.3 is complete.

Proposition 7.3 completes the proof of Theorem 7.1.

CHAPTER 3

Isotropy of virtual Albert forms over function fields ofquadrics

Let F be a field of characteristic different from 2 and ϕ be a virtual Albertform over F , i.e. an anisotropic 6-dimensional quadratic form over F whichis still anisotropic over the field F (

√d± ϕ). We give a complete description

of the quadratic forms ψ such that ϕ becomes isotropic over the function fieldF (ψ). This completes the series of works ([16], [44], [45], [49], [54]) wherethe question was considered previously.

Results of this Chapter are obtained in joint work with Oleg Izhboldin.

0. Introduction

Let F be a field of characteristic different from 2 and let ϕ and ψ be twoanisotropic quadratic forms over F . An important problem in the algebraictheory of quadratic forms is to find conditions on ϕ and ψ so that ϕF (ψ) isisotropic. In the case when dimϕ ≤ 5 the problem was completely solved in[15] and [75]. For 6-dimensional quadratic forms, the problem was studied byD. W. Hoffmann ([16]), A. Laghribi ([44], [45]), D. Leep ([49]), and A. S.Merkurjev ([54]) and was solved fully except for the following two cases (see[44] and [45]):

• dimψ = 4, d± ψ = 1, and ind(C0(ϕ)) = 2;• dimψ = 4, d± ψ = 1, ind(C0(ϕ)) = 4, and d± ϕ = d± ψ.

In this Chapter the second case is studied completely. Our result (Theorem5.1) and results of Laghribi, Leep, and Merkurjev give rise to the following

Theorem. Let ϕ be a 6-dimensional quadratic form with ind(C0(ϕ)) = 4.In the case when ψ /∈ GP2(F ), the quadratic form ϕF (ψ) is isotropic if andonly if ψ is similar to a subform of ϕ. In the case when ψ ∈ GP2(F ), the formϕF (ψ) is isotropic if and only if a 3-dimensional subform of ψ is similar to asubform of ϕ.

We deduce Theorem 5.1 from a result on 8-dimensional forms (Proposition4.1) which also has an independent value: together with [47], it gives riseto Theorem 4.4 answering the question about isotropy of an 8-dimensionalquadratic form ϕ with detϕ = 1 and ind(C(ϕ)) = 8 over function fields ofquadrics.

49

50 3. VIRTUAL ALBERT FORMS

1. Terminology, notation, and backgrounds

1.1. Quadratic forms. By ϕ ⊥ ψ, ϕ ≃ ψ, and [ϕ] we denote orthogonalsum of forms, isometry of forms, and the class of ϕ in the Witt ring W (F ) ofthe field F respectively. To simplify notation we will write ϕ+ψ instead of [ϕ]+[ψ]. For a quadratic form ϕ of dimension n, we set d± ϕ = (−1)n(n−1)/2 detϕ.We consider d± ϕ as an element of F ∗/F ∗2. The maximal ideal of W (F )consisting of the classes of the even-dimensional forms is denoted by I(F ).The anisotropic part of ϕ is denoted by ϕan. We denote by ⟨⟨a1, . . . , an⟩⟩ then-fold Pfister form

⟨1,−a1⟩ ⊗ · · · ⊗ ⟨1,−an⟩and by Pn(F ) the set of all n-fold Pfister forms. The set of all forms similar ton-fold Pfister forms we denote by GPn(F ). For any field extension L/F , we putϕL = ϕ ⊗ L, W (L/F ) = Ker(W (F ) → W (L)), and In(L/F ) = Ker(In(F ) →In(L)).

For a quadratic form ϕ of dimension ≥ 3, we denote by Xϕ the projectivevariety given by the equation ϕ = 0. We set F (ϕ) = F (Xϕ) if dimϕ ≥ 3;

F (ϕ) = F (√d) if dimϕ = 2 and d = d± ϕ = 1; and F (ϕ) = F otherwise.

Let ψ ∈ GP2(F ) and ψ0 be a 3-dimensional subform of ψ. Then ψF (ψ0) and(ψ0)F (ψ) are isotropic. Hence for any quadratic form ϕ, the isotropy of ϕF (ψ)

is equivalent to isotropy of ϕF (ψ0). Thus, to give a complete description of thequadratic forms ψ such that ϕ becomes isotropic over the function field F (ψ),it is sufficient to consider the case where ψ /∈ GP2(F ).

We say that a quadratic form ϕ is a Pfister neighbor if for some n thereexists π ∈ Pn(F ) such that ϕ is similar to a subform of π and dimϕ > 2n−1.

Let ϕ be a quadratic form of dimension 2n. We say that ϕ∗ is a half-neighborof ϕ, if dimϕ∗ = 2n and there exists k ∈ F ∗ such that ϕ∗ ≡ kϕ (mod In+1(F )).

1.2. Algebras. Let A be a central simple algebra over F . By deg(A),ind(A), [A], and exp(A) we denote respectively the degree of A, the Schurindex of A, the class of A in the Brauer group Br(F ), and the order of [A] inthe Brauer group. By SB(A) we denote the Severi–Brauer variety of A. If analgebra B has the form B = A× A, we set indB = indA.

Let ϕ be a quadratic form. We denote by C(ϕ) the Clifford algebra of ϕ.By C0(ϕ) we denote the even part of C(ϕ). If ϕ ∈ I2(F ) then C(ϕ) is a centralsimple algebra. Hence we get a well defined element [C(ϕ)] of Br2(F ) whichwe denote by c(ϕ).

1.3. Quadratic forms of dimension 6. Let ϕ be an anisotropic qua-dratic form of dimension 6 and let d = d± ϕ. If d = 1, then ϕ is an Albertform. In this case the problem of isotropy of ϕ over the function field of a qua-dratic form ψ is completely solved ([49], [54]): in the case when ψ /∈ GP2(ψ),the form ϕF (ψ) is isotropic if and only if ψ is similar to a subform in ϕ.

Suppose now that d = 1. Then C0(ϕ) is a central simple algebra over

L = F (√d). In this case we have the following classification of anisotropic

6-dimension forms:

1. TERMINOLOGY, NOTATION, AND BACKGROUNDS 51

Type 1 is defined by one of the following equivalent conditions:

• ind(C0(ϕ)) = 1;• ϕL is hyperbolic;• ϕ has the form ⟨⟨d⟩⟩µ where dimµ = 3;• ϕ is a Pfister neighbor.


• ind(C0(ϕ)) = 2;• ϕL is isotropic but not hyperbolic;• ϕ is similar to a form of the kind ⟨⟨a, b⟩⟩ ⊥ c ⟨⟨d⟩⟩, where ⟨⟨a, b⟩⟩L is notisotropic.


• ind(C0(ϕ)) = 4;• ϕL is anisotropic;

The quadratic form of the type 3 is called a virtual Albert form.

For the quadratic forms ϕ of type 1 (i.e. for the Pfister neighbors), theproblem of isotropy ϕF (ψ) is completely solved by Arason–Pfister subform the-orem. The case of quadratic forms of type 2 was studied by D. Hoffmann in[16]: he found the conditions on ϕ and ψ so that ϕF (ψ) is isotropic exceptingthe case dimψ = 4. 1

The case of the quadratic forms ϕ of type 3 (virtual Albert forms) wasstudied completely by A. Laghribi in [44, 45] except for the case where dimψ =4 and d± ψ = 1. In this chapter we complete the investigation of isotropy ofvirtual Albert forms over the function field of a quadric.

1.4. Cohomology groups. By H∗(F ) we denote the graded ring of Ga-lois cohomology

H∗(F,Z/2Z) = H∗(Gal(Fsep/F),Z/2Z).

For any field extension L/F , we set H∗(L/F ) = Ker(H∗(F ) → H∗(L)).We use the standard canonical isomorphisms H0(F ) = Z/2Z, H1(F ) =

F ∗/F ∗2, and H2(F ) = Br2(F ). Thus any element a ∈ F ∗ gives rise to anelement of H1(F ); it is denoted by (a). The cup product (a1) ∪ · · · ∪ (an) isdenoted by (a1, . . . , an).

For n = 0, 1, 2 there is a homomorphism en : In(F ) → Hn(F ) definedas follows: e0(ϕ) = dimϕ (mod 2), e1(ϕ) = d± ϕ, and e2(ϕ) = c(ϕ). More-over, there exists a homomorphism e3 : I3(F ) → H3(F ) which is uniquelydetermined by the condition e3(⟨⟨a1, a2, a3⟩⟩) = (a1, a2, a3) (see [5]). The ho-momorphism en is surjective and Ker en = In+1(F ) for n = 0, 1, 2, 3 (see [53],[62], and [71]).

1The case where ϕ is of type 2 and dimψ = 4 is studied in Chapter 4.


1.5. K-theory and Chow groups. In §2 we use the following notation.Let X be a smooth algebraic F -variety. The Grothendieck ring of X is

denoted by K(X). This ring is supplied with the filtration “by codimensionof support” (which respects the multiplication); the adjoint graded ring isdenoted by G∗K(X). There is a canonical surjective homomorphism of thegraded Chow ring CH∗(X) onto G∗K(X); its kernel consists only of torsionelements and is trivial in the 0-th, 1-st and 2-nd graded components ([81, §9]).

We fix a separable closure F of the ground field F and denote by X thevariety XF . The image of the restriction homomorphism G∗K(X) → G∗K(X)is denoted by G∗K(X).

We denote by |S| the order of a finite set S.

2. Computation of H3(F (SB(A)× SB(B))/F )

Theorem 2.1. Let A and B be biquaternion division F -algebras such thatind(A⊗B) = 8. Suppose that there exists a quadratic extension L/F such thatAL and BL are not division algebras. Then

H3(F (SB(A)× SB(B))/F ) = [A] ·H1(F ) + [B] ·H1(F ).

Proof. We put X = SB(A)× SB(B).

Lemma 2.2 ([33, Proposition 2]).

|TorsG∗K(X)| = |G∗K(X)/G∗K(X)||K(X)/K(X)|

.

Lemma 2.3. |K(X)/K(X)| = 228.

Proof. Applying [69, §8, Theorem 4.1], one gets an isomorphism

K(X) ≃ K(F )⊕4 ⊕K(A)⊕4 ⊕K(B)⊕4 ⊕K(A⊗B)⊕4

which shows that

|K(X)/K(X)| = (indA)4 · (indB)4 · (indA⊗B)4 = 228 .

The variety SB(A) is a projective space; denote by f the class of a hyper-

plane in G1K(SB(A)).

Lemma 2.4. For any i ≥ 0, the group GiK(SB(AL)) contains

• f i, if i is even,• 2f i, if i is odd.

Proof. By [33, Lemma 3], for any i, one has an inclusion

GiK(SB(AL)) ∋indAL

(i, indAL)f i

where (· , ·) denotes the greatest common divisor. Since indAL = 2, the state-ment follows.

2. COMPUTATION OF H3(F (SB(A)× SB(B))/F ) 53

Lemma 2.5. G1K(SB(A)) ∋ 2f .

Proof. By the computation [6, §2] of the Picard group of a Severi-Brauervariety, one knows that

G1K(SB(A)) ∋ (expA)f .

Since expA = 2, the statement follows.

The variety SB(B) is a projective space; denote by g the class of a hyper-

plane in G1K(SB(B)).

Corollary 2.6. For any i, j ≥ 0, the group Gi+jK(X) contains

• f i × gj, if i = j = 0;• 2(f i × gj), if

i and j are even ori = 0, j = 1 ori = 1, j = 0;

• 4(f i × gj), ifi+ j is odd ori = j = 1;

• 8(f i × gj) for any i, j.

Proof. The case i = j = 0 is evident.If i and j are even, then f i ∈ GiK(SB(AL)) and gj ∈ GjK(SB(BL)) by

Lemma 2.4. Thus f i× gj ∈ Gi+jK(XL) and the transfer argument shows that2(f i × gj) ∈ Gi+jK(X).

By Lemma 2.5, G1K(SB(A)) ∋ 2f and G1K(SB(B)) ∋ 2g. ThereforeG1K(X) contains 2(f × 1) and 2(1× g); moreover, G2K(X) ∋ 4(f × g).

If i+j is odd, then 2(f i×gj) ∈ Gi+jK(XL) by Lemma 2.4 and the transferargument shows that 4(f i × gj) ∈ Gi+jK(X).

Since there exists a field extension of degree 8 simultaneously splitting thealgebras A and B, the inclusion 8(f i × gj) ∈ Gi+jK(X) holds for any i, j.

Corollary 2.7. |G∗K(X)/G∗K(X)| ≤ 228.

Proof. Since SB(A) and SB(B) are projective spaces, G∗K(X) is an abeliangroup freely generated by f i×gj with i, j = 0, 1, 2, 3. By Lemma 2.6, we knowthat the following multiples of these generators are in G∗K(X):

20 · (f0 × g0), 21 · (f0 × g1), 21 · (f 0 × g2), 22 · (f 0 × g3),21 · (f1 × g0), 22 · (f1 × g1), 22 · (f 1 × g2), 23 · (f 1 × g3),21 · (f2 × g0), 22 · (f2 × g1), 21 · (f 2 × g2), 22 · (f 2 × g3),22 · (f3 × g0), 23 · (f3 × g1), 22 · (f 3 × g2), 23 · (f 3 × g3).

Taking the product of the coefficients, we get 228.

Corollary 2.8. TorsG∗K(X) = 0.

Proof. Follows from Lemma 2.2, Lemma 2.3 and Corollary 2.7.


Since the Chow group CH2(X) is isomorphic to G2K(X) ([81, §9]), we alsoget

Corollary 2.9. TorsCH2(X) = 0.

To complete the proof of Theorem 2.1, we apply [66, Theorem 4.1 withRemark 4.1]. By that result, there is a monomorphism

H3(F (X)/F )

[A] ·H1(F ) + [B] ·H1(F )↪→ TorsCH2(X)

and so, by Corollary 2.9, we are done.

3. Computation of H3(F (Xψ × SB(D))/F )

Theorem 3.1. Let ψ = ⟨−a,−b, ab, d⟩ be an anisotropic quadratic formover F with d /∈ F ∗2. Let D = (a, b)⊗ (u, v)⊗ (d, s) be a division 3-quaternionalgebra over F . Then the group

H3(F (Xψ × SB(D))/F )

is equal to

{e3(ψ ⟨⟨k⟩⟩) | k ∈ F ∗ is such that ψ ⟨⟨k⟩⟩ ∈ GP3(F )}+ [D]H1(F ).

Proof. Put F = F ((t)). Consider two biquaternion algebras A = (a, b)⊗(d, t) and B = (d, st)⊗ (u, v) over F .

Since D is a division algebra, it follows that ind(D) = 8. Therefore

ind((a, b)⊗ (u, v))F (√d) = indDF (

√d) = 4 .

Hence (a, b)F (√d) and (u, v)F (

√d) are division F (

√d)-algebras. By Tignol’s

theorem [83, Proposition 2.4], A and B are division F -algebras as well.

Since [A⊗B] = [DF ] in Br(F ), we have ind(A⊗F B) = ind(D) = 8. SinceAF (

√d) and BF (

√d) are not division algebras, the conditions of Theorem 2.1

hold for the field F and algebras A and B over F . Therefore

H3(F (SB(A)× SB(B))/F ) = [A]H1(F ) + [B]H1(F ).

Let E = F (SB(A) × SB(B)). Clearly [AE] = [BE] = 0. Hence [DE] =[AE] + [BE] = 0. Thus SB(D)E is a rational variety.

Since [AE] = 0, the Albert form ⟨−a,−b, ab, d, t,−dt⟩E of the biquater-nion algebra AE = ((a, b) ⊗ (d, t))E is hyperbolic. Hence ⟨−a,−b, ab, d⟩E =⟨dt,−d⟩E in the Witt ring W (E). Therefore ψE is isotropic. Hence (Xψ)E isa rational variety.

Let Y = Xψ×SB(D). Since (Xψ)E and SB(D)E are rational, it follows thatYE is rational. Hence E(Y )/E is a purely transcendental extension. There-fore H3(E(Y )/F ) = H3(E/F ). We have H3(F (Y )/F ) ⊂ H3(E(Y )/F ) =H3(E/F ).

3. COMPUTATION OF H3(F (Xψ × SB(D))/F ) 55

Let u ∈ H3(F (Xψ × SB(D))/F ) = H3(F (Y )/F ). To prove the theorem, itis enough to show that u can be written in the form

u = e3(ψ ⟨⟨k⟩⟩) + [D] ∪ (r)

with some k, r ∈ F ∗.Since H3(F (Y )/F ) ⊂ H3(E/F ), it follows that

uF ∈ H3(E/F ) = H3(F (SB(A)× SB(B))/F ) = [A]H1(F ) + [B]H1(F ).

Since [A]+ [B] = [DF ], we have uF ∈ [A]H1(F )+ [DF ]H1(F ). Hence there are

α, β ∈ F such that uF = [A]∪(α)+[DF ]∪(β). Since F ∗/F ∗2 ≃ F ∗/F ∗2×{1, t},we can suppose that α = kti and β = rtj, where k, r ∈ F ∗ and i, j ∈ {0, 1}.We have

uF = [A] ∪ (α) + [DF ] ∪ (β) == ((a, b) + (d, t)) ∪ (kti) + [D] ∪ (rtj) == ((a, b, k) + [D] ∪ (r)) + (t) ∪ (i(a, b) + (d, k(−1)i) + j[D]).

Using well-known isomorphism H i(F ((t))) = H i(F )⊕H i−1(F ), we have

u = (a, b, k) + [D] ∪ (r)

andi(a, b) + (d, k(−1)i) + j[D] = 0.

We claim that j = 0. Indeed, if j = 0 then j = 1 and hence [D] =i(a, b) + (d, k(−1)i). Therefore [D] = [(a, bi)⊗ (d, k(−1)i)]. Thus ind(D) ≤ 4,a contradiction.

So j = 0, and we have i(a, b) + (d, k(−1)i) = 0. Thereby i(a, b)F (√d) = 0.

Since (a, b)F (√d) = 0, it follows that i = 0.

Since i(a, b) + (d, k(−1)i) = 0 and i = 0, we have (d, k) = 0. Hence⟨⟨d, k⟩⟩ = 0 in W (F ). Since ψ = ⟨−a,−b, ab, d⟩ = ⟨⟨a, b⟩⟩ − ⟨⟨d⟩⟩, we have

ψ ⟨⟨k⟩⟩ = (⟨⟨a, b⟩⟩ − ⟨⟨d⟩⟩) ⟨⟨k⟩⟩ = ⟨⟨a, b, k⟩⟩ − ⟨⟨d, k⟩⟩ = ⟨⟨a, b, k⟩⟩ .Therefore ψ ⟨⟨k⟩⟩ ∈ GP3(F ) and e

3(ψ ⟨⟨k⟩⟩) = (a, b, k).Hence the element u = (a, b, k) + [D] ∪ (r) belongs to the set

{e3(ψ ⟨⟨k⟩⟩) | k is such that ψ ⟨⟨k⟩⟩ ∈ GP3(F )}+ [D]H1(F ).

Proposition 3.2. In the notation of Theorem 3.1, let ξ ∈ I2(F ) be aquadratic form such that c(ξ) = [D]. Then for an arbitrary element π ∈I3(F (Xψ × SB(D))/F ) there are k1, k2 ∈ F ∗ such that

π ≡ ψ ⟨⟨k1⟩⟩+ ξ ⟨⟨k2⟩⟩ (mod I4(F ))

Proof. Obviously e3(π) ∈ H3(F (Xψ × SB(D))/F ). It follows from The-orem 3.1 that there are k1, k2 ∈ F ∗ such that e3(π) = e3(ψ ⟨⟨k1⟩⟩) + [D] ∪(k2). Clearly [D] ∪ (k2) = e2(ξ) ∪ e1(⟨⟨k2⟩⟩) = e3(ξ ⟨⟨k2⟩⟩). Hence e3(π) =e3(ψ ⟨⟨k1⟩⟩) + e3(ξ ⟨⟨k2⟩⟩). Since Ker(e3 : I3(F ) → H3(F )) = I4(F ), we haveπ ≡ ψ ⟨⟨k1⟩⟩+ ξ ⟨⟨k2⟩⟩ (mod I4(F )).


4. 8-dimensional quadratic forms

Proposition 4.1. Let ϕ be an 8-dimensional quadratic form with d± ϕ =1 and indC(ϕ) = 8. Let ψ be a 4-dimensional quadratic form with d± ψ = 1.Suppose that ϕF (ψ) is isotropic. Then there exists a half-neighbor ϕ∗ of ϕ suchthat ψ ⊂ ϕ∗.

Proof. Changing ψ by a coefficient we can assume that ψ = ⟨−a,−b, ab, d⟩with d /∈ F ∗2. Clearly C0(ψ) = (a, b)F (

√d). Since indC(ϕ) = 8, there exists a

3-quaternion division algebra D such that c(ϕ) = [D].

Lemma 4.2. There exist u, v, s ∈ F ∗ such that D = (a, b)⊗ (u, v)⊗ (d, s).

Proof. Since ϕF (ψ) is isotropic, it follows that DF (ψ) is not a divisionalgebra. The index reduction formula [55] shows that ind(C0(ψ) ⊗F D) = 2.Therefore ind((a, b) ⊗F D)F (

√d) = 2. Hence there are u, v, s ∈ F ∗ such that

[(a, b)⊗FD] = [(u, v)⊗F (d, s)] in Br2(F ). Hence [D] = [(a, b)⊗F (u, v)⊗F (d, s)].Since degD = 8, we have D = (a, b)⊗ (u, v)⊗ (d, s).

Consider the quadratic form

γ = ⟨−a,−b, ab, d⟩ ⊥ −s ⟨−u,−v, uv, d⟩ .One can verify that d± γ = 1 and c(γ) = [D]. Hence c(ϕ) = [D] = c(γ).Therefore ϕ+ γ ∈ I3(F ).

Lemma 4.3. ϕ+ γ ∈ I3(F (Xψ × SB(D))/F ).

Proof. Let E = F (Xψ × SB(D)). Since ϕ+ γ ∈ I3(F ), it is sufficient toverify that ϕE and γE are hyperbolic. Obviously [DE] = 0, and the form ψEis isotropic. Since c(ϕE) = c(γE) = [DE] = 0 and dimϕ = dim γ = 8, we haveϕE, γE ∈ GP3(E). Hence it is sufficient to prove that ϕE and γE are isotropic.Since ϕF (ψ) and ψE are isotropic, ϕE is isotropic as well. Since ψ ⊂ γ and ψEis isotropic, we see that γE is isotropic.

Now we can complete the proof of Proposition 4.1. By Proposition 3.2 andLemma 4.3, there exist k1, k2 ∈ F ∗ such that

ϕ+ γ ≡ ψ ⟨⟨k1⟩⟩+ ϕ ⟨⟨k2⟩⟩ (mod I4(F )).

Let ρ = −s ⟨−u,−v, uv, d⟩. We have γ = ψ + ρ. Hence

ϕ+ ψ + ρ ≡ ψ − k1ψ + ϕ− k2ϕ (mod I4(F )).

Thus k1ψ + ρ ≡ −k2ϕ (mod I4(F )). Hence ψ + k1ρ ≡ −k1k2ϕ (mod I4(F )).We finish the proof by setting ϕ∗ = ψ ⊥ k1ρ.

Theorem 4.4. Let ϕ be an 8-dimensional quadratic form with d± ϕ = 1and indC(ϕ) = 8. Let ψ be a quadratic form of dimension ≥ 4 such thatψ /∈ GP2(F ). The following conditions are equivalent:

• ϕF (ψ) is isotropic;• there exists a half-neighbor ϕ∗ of ϕ such that ψ ⊂ ϕ∗.

5. MAIN THEOREM 57

Proof. The case dimψ = 4 is Proposition 4.1. In the case dimψ = 4 thestatement was proved by Laghribi in [47] and [45].

5. Main theorem

Theorem 5.1. Let ϕ be a virtual Albert form (i.e. a 6-dimensional qua-dratic form with d± ϕ /∈ F ∗2 and ind(C0(ϕ)) = 4). Let ψ be a 4-dimensionalquadratic form such that d± ψ = 1. The following conditions are equivalent:

(1) ϕF (ψ) is isotropic;(2) ψ is similar to a subform in ϕ.

Proof. (1)⇒(2). Let d = d± ϕ. Consider the 8-dimensional quadratic

form ξ = ϕF ⊥ t ⟨⟨d⟩⟩ over the field F = F ((t)). Since d± ϕ /∈ F ∗2 andind(C0(ϕ)) = 4, one has ind(C(ξ)) = 8. Clearly ξF (ψ) is isotropic. It follows

from Proposition 4.1 that there exists a quadratic form ξ∗ over F such that ξand ξ∗ are half-neighbors and ψF ⊂ ξ∗.

Lemma 5.2. ξ∗ is similar to ξ.

Proof. Since ξ and ξ∗ are half-neighbors, there exists k ∈ F such that

ξ ≡ kξ∗ (mod I4(F )). By Springer’s theorem one can write kξ∗ in the formkξ∗ = µ0 ⊥ tµ1, where quadratic forms µ0 and µ1 are defined over F . We have

ϕ ⊥ t ⟨⟨d⟩⟩ = ξ ≡ kξ∗ = µ0 ⊥ tµ1 (mod I4(F )).

Hence ϕ ≡ µ0 (mod I3(F )), ⟨⟨d⟩⟩ ≡ µ1 (mod I3(F )), and ϕ + ⟨⟨d⟩⟩ ≡ µ0 + µ1

(mod I4(F )). Therefore indC0(µ0) = indC0(ϕ) ≥ 4. Hence dimµ0 ≥ 6.Therefore dimµ1 ≤ 2. By Arason–Pfister Hauptsatz the condition ⟨⟨d⟩⟩ ≡ µ1

(mod I3(F )) implies that µ1 = ⟨⟨d⟩⟩. Hence ϕ ≡ µ0 (mod I4(F )). ApplyingArason–Pfister Hauptsatz once again, we have ϕ = µ0. Therefore ξ = kξ∗.

Now we return to the proof of Theorem 5.1. Since ψ is similar to a subformin ξ∗, and ξ∗ is similar to ξ, it follows that ψ is similar to a subform in ξ = ϕ ⊥t ⟨⟨d⟩⟩. Thus ψ is similar to a subform of ϕ by the following obvious observation.

Lemma 5.3. Let ψ, γ0 and γ1 be anisotropic quadratic forms over F . Thefollowing conditions are equivalent:

a) ψF ((t)) is similar to a subform in γ0 ⊥ tγ1,b) ψ is similar either to a subform in γ0 or to a subform in γ1.

Thus we have proved that condition (1) of Theorem 5.1 implies condition(2). On the other hand, condition (2) obviously implies condition (1). Theproof of Theorem 5.1 is complete.

Theorem 5.4. Let ϕ be a virtual Albert form and let ψ /∈ GP2(F ). Thequadratic form ϕF (ψ) is isotropic if and only if ψ is similar to a subform in ϕ.

Proof. This theorem was proved by A. Laghribi in the following cases([44], [45]):

• dimψ = 4;


• dimψ = 4, d± ψ = d± ϕ.

Thus we can suppose that dimψ = 4. To complete the proof it is sufficient toapply Theorem 5.1.

In the special case which was not covered by the results of A. Laghribi,we get the following

Corollary 5.5. Let ϕ be a virtual Albert form and ψ be a 4-dimensionalform such that d± ψ = d± ϕ. Then ϕF (ψ) is anisotropic.

Proof. If ψ is similar to a subform in ϕ, then ϕ is isotropic, a contradic-tion. Therefore ψ is not similar to a subform in ϕ. By Theorem 5.1, it meansthat ϕF (ψ) is anisotropic.

Together with results described in §1, Theorem 5.4 gives rise to the follow-ing

Corollary 5.6. Let ϕ be a 6-dimensional quadratic form with indC0(ϕ) =4. In the case when ψ /∈ GP2(F ), the quadratic form ϕF (ψ) is isotropic if andonly if ψ is similar to a subform of ϕ. In the case when ψ ∈ GP2(F ), the formϕF (ψ) is isotropic if and only if a 3-dimensional subform of ψ is similar to asubform of ϕ.

CHAPTER 4

Isotropy of 6-dimensional quadratic forms over functionfields of quadrics

Let F be a field of characteristic different from 2 and ϕ be an anisotropic6-dimensional quadratic form over F . We study the last open cases in theproblem of describing the quadratic forms ψ such that ϕ becomes isotropicover the function field F (ψ).


0. Introduction

Let F be a field of characteristic different from 2 and let ϕ and ψ be twoanisotropic quadratic forms over F . An important problem in the algebraictheory of quadratic forms is to find conditions on ϕ and ψ so that ϕF (ψ) isisotropic.

More precisely, one studies the question whether the isotropy of ϕ over F (ψ)is standard in a sense. In this chapter we will use the following definition of“standard isotropy”:

Definition. Let ϕ and ψ be anisotropic quadratic forms such that ϕF (ψ) isisotropic. We say that the isotropy of ϕF (ψ) is standard, if at least one of thefollowing conditions holds:

• ψ is similar to a subform in ϕ;• there exists a subform ϕ0 ⊂ ϕ with the following two properties:

– the form ϕ0 is a Pfister neighbor,– the form (ϕ0)F (ψ) is isotropic.

Otherwise, we say that the isotropy is non-standard.

In the case when dimϕ ≤ 5, the isotropy of ϕF (ψ) is always standard ([75],[15]). For 6-dimensional quadratic forms, the problem was studied by A. S.Merkurjev ([54]), D. Leep ([49]), D. W. Hoffmann ([16]), A. Laghribi ([44],[45]), and in Chapter 3. It was proved that isotropy of a 6-dimensional qua-dratic form ϕ over the function field of a quadratic form ψ is always standardexcept (possibly) the following case (see [44] and Chapter 3):

• dimψ = 4, d± ψ = 1, d± ϕ = 1, and indC0(ϕ) = 2.

In this Chapter we study the isotropy of ϕF (ψ) for quadratic forms ϕ and ψsatisfying these conditions (with dimϕ = 6).

Note that the condition indC0(ϕ) = 2 implies that there exist a, b, c, d ∈ F ∗

such that ϕ is similar to the form ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩. Since ϕ can be replacedby a similar form, we can assume that ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩. Note that

59

60 4. ISOTROPY OF 6-DIMENSIONAL QUADRATIC FORMS

in this case [C0(ϕ)] = [(a, b)F (√d)] = [C0(ρ)], where ρ is defined as follows:

ρ = ⟨−a,−b, ab, d⟩.Since dimψ = 4, there exist u, v, δ ∈ F ∗ such that ψ is similar to the

quadratic form ⟨−u,−v, uv, δ⟩. Since d± ψ = 1, we have δ /∈ F ∗2. Thus ourmain problem is reduced to the following

Question. Let ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩ and ψ = ⟨−u,−v, uv, δ⟩ be anisotropicquadratic forms over F with d, δ /∈ F ∗2. Suppose that ϕF (ψ) is isotropic. Is theisotropy standard?

This question naturally splits into the following four cases:

(1) d = δ as elements of F ∗/F ∗2,(2) d = δ and indC0(ϕ)⊗F C0(ψ) = 1,(3) d = δ and indC0(ϕ)⊗F C0(ψ) = 2,(4) d = δ and indC0(ϕ)⊗F C0(ψ) = 4.

We prove that in the cases (1), (2), and (4) the isotropy of ϕF (ψ) is alwaysstandard (see Theorem 8.5, Propositions 8.6 and 8.7). This statement givesrise to the following

Theorem. Let ϕ be an anisotropic quadratic form of dimension ≤ 6 and ψbe such that ϕF (ψ) is isotropic. Then isotropy is standard except (possibly) thefollowing case: dimϕ = 6, dimψ = 4, 1 = d± ϕ = d± ψ = 1, and indC0(ϕ) =2 = indC0(ϕ)⊗F C0(ψ).

The proof of this theorem is based on a computation of the second Chowgroup for certain homogeneous varieties. Namely, we show that the ques-tion on the standard isotropy can be reduced to a question on the groupTorsCH2(Xψ×Xρ), where ρ = ⟨−a,−b, ab, d⟩ and Xψ and Xρ are the pro-jective quadrics corresponding to ψ and ρ. In the cases (1), (2) and (4), wecompute the group TorsCH2(Xψ×Xρ) completely (see Theorems 5.7, 5.1, 5.8,and Lemma 7.7):

Theorem. Let ψ and ρ be 4-dimensional quadratic forms. Then the groupTorsCH2(Xψ ×Xρ) is zero or isomorphic to Z/2Z. Moreover,

• if detψ = det ρ or if indC0(ψ)⊗F C0(ρ) = 4, then the group CH2(Xψ ×Xρ) is torsion-free;

• in the case indC0(ψ)⊗F C0(ρ) = 1, the group CH2(Xψ×Xρ) is torsion-free if and only if ρ and ψ contain similar 3-dimensional subforms.

It will be shown later (in Chapter 5) that in the case (3), i.e. in the casewhere d = δ and indC0(ϕ) ⊗F C0(ψ) = 2, the answer to our main question(whether the isotropy of ϕF (ψ) is always standard) is negative. Here we showthat this question is equivalent to the following one (see §9): is the groupTorsCH2(Xψ × Xρ) trivial for any 4-dimensional quadratic forms ψ and ρsuch that 1 = detψ = det δ = 1 and indC0(ψ)⊗ C0(ρ) = 2?



Quadratic forms. By ϕ ⊥ ψ, ϕ ≃ ψ, and [ϕ] we denote respectivelyorthogonal sum of forms, isometry of forms, and the class of ϕ in the Witt ringW (F ) of the field F . To simplify notation, we write ϕ1+ϕ2 instead of [ϕ1]+[ϕ2].For a quadratic form ϕ of dimension n, we set d± ϕ = (−1)n(n−1)/2 detϕ ∈F ∗/F ∗2. The maximal ideal of W (F ) generated by the classes of the even-dimensional forms is denoted by I(F ). The anisotropic part of ϕ is denotedby ϕan. We denote by ⟨⟨a1, . . . , an⟩⟩ the n-fold Pfister form

⟨1,−a1⟩ ⊗ · · · ⊗ ⟨1,−an⟩

and by Pn(F ) the set of all n-fold Pfister forms. The set of all forms similarto an n-fold Pfister form we denote by GPn(F ). For any field extension L/F ,we put ϕL = ϕ ⊗F L, W (L/F ) = ker(W (F ) → W (L)), and In(L/F ) =ker(In(F ) → In(L)).

For a quadratic form ϕ and a field extension L/F , we denote by DL(ϕ) theset of the non-zero values of the quadratic form ϕL.

For a quadratic form ϕ of dimension ≥ 3, we denote by Xϕ the projectivevariety given by the equation ϕ = 0. We set F (ϕ) = F (Xϕ) and F (ϕ, ψ) =F (Xϕ ×Xψ) for quadratic forms ϕ and ψ of dimensions ≥ 3.

Algebras. We consider only finite-dimensional F -algebras.For a simple F -algebra A, by ind(A) we denote the Schur index of A. For

an algebra B of the form B = A×· · ·×A with simple A, we set indB = indA.Let ϕ be a quadratic form. We denote by C(ϕ) the Clifford algebra of ϕ.

By C0(ϕ) we denote the even part of C(ϕ). For any collection ρ1, . . . , ρm ofquadratic forms, the algebra C0(ρ1)⊗F · · ·⊗F C0(ρm) is of the form A×· · ·×Awith simple A. Therefore, we get a well-defined positive integer indC0(ρ1)⊗F

· · · ⊗F C0(ρm).If ϕ ∈ I2(F ) then C(ϕ) is a central simple algebra. Hence we get a well-

defined element [C(ϕ)] in the 2-part Br2(F ) of the Brauer group Br(F ) whichwe denote by c(ϕ).

Cohomology groups. By H∗(F ) we denote the graded ring of Galois

cohomology H∗(F,Z/2Z) def= H∗(Gal(Fsep/F ),Z/2Z). For any field extension

L/F , we set H∗(L/F ) = ker(H∗(F ) → H∗(L)

).

We use the standard canonical isomorphisms H0(F ) = Z/2Z, H1(F ) =F ∗/F ∗2, and H2(F ) = Br2(F ). So any element a ∈ F ∗ gives rise to an elementof H1(F ) which we denote by (a). The cup product (a1)∪ · · ·∪ (an) we denoteby (a1, . . . , an).

For n = 0, 1, 2, there is a homomorphism en : In(F ) → Hn(F ) definedas follows: e0(ϕ) = dimϕ (mod 2), e1(ϕ) = d± ϕ, and e2(ϕ) = c(ϕ). More-over there exists a homomorphism e3 : I3(F ) → H3(F ) which is uniquelydetermined by the condition en(⟨⟨a1, a2, a3⟩⟩) = (a1, a2, a3) (see [5]). The ho-momorphism en is surjective and ker en = In+1(F ) for n = 0, 1, 2, 3 (see [53],[62], and [71]).


We also work with the cohomology groups Hn(F,Q/Z(i)), (i = 0, 1, 2),defined by B. Kahn (see [29]). For any field extension L/F , we set

H∗(L/F,Q/Z(i)) = ker(H∗(F,Q/Z(i)) → H∗(L,Q/Z(i))

).

For n = 1, 2, 3, the group Hn(F ) is naturally identified with the 2-part ofHn(F,Q/Z(n− 1)).

K-theory and Chow groups. For a smooth algebraic F -variety X, itsGrothendieck ring is denoted byK(X). This ring is supplied with the filtrationby codimension of support (which respects the multiplication). For a ring (or agroup) with filtration A, we denote by G∗A the adjoint graded ring (resp., theadjoint graded group). There is a canonical surjective homomorphism of thegraded Chow ring CH∗(X) onto G∗K(X), its kernel consists only of torsionelements and is trivial in the 0-th, 1-st, and 2-nd graded components ([81,§9]).

2. The group H3(F (ρ1, ρ2)/F )

The main result of this § (in view of our further purposes) is Corollary 2.13.By a homogeneous variety we always mean a projective homogeneous vari-

ety.

Proposition 2.1 ([67]). For any homogeneous F -variety X, there is anatural (in X and in F ) epimorphism

τX : H3(F (X)/F,Q/Z(2)

)� TorsCH2(X) .

Proposition 2.2. For any homogeneous varieties X1, . . . , Xm over F , thequotient

H3(F (X1 × · · · ×Xm)/F,Q/Z(2)

)H3(F (X1)/F,Q/Z(2)

)+ · · ·+H3

(F (Xm)/F,Q/Z(2)

)is canonically isomorphic to

TorsCH2(X1 × · · · ×Xm)

pr∗1TorsCH2(X1) + · · ·+ pr∗mTorsCH2(Xm)

where pr∗1, . . . , pr∗m are the pull-backs with respect to the projections pr1, . . . , prm

of the product X1 × · · · ×Xm on X1, . . . , Xm.

Proof. Set X = X1 × · · · × Xm. The homomorphism τX of Proposition2.1 induces an epimorphism

f :H3(F (X1 × · · · ×Xm)/F,Q/Z(2)

)H3(F (X1)/F,Q/Z(2)

)+ · · ·+H3

(F (Xm)/F,Q/Z(2)

) �

� TorsCH2(X1 × · · · ×Xm)


with the kernel ker f = ker τX/(ker τX1 + · · ·+ ker τXm).The kernel of τX is computed (for any homogeneous X) in [57]: let A

be the separable F -algebra associated with X ([57, §2]) and denote by E the

2. THE GROUP H3(F (ρ1, ρ2)/F ) 63

center of A; then ker τX = {NE/F (x∪ [A]) | with x ∈ E∗} where [A] is the classof A in the Brauer group Br(E) = H2(E,Q/Z(1)), x is the class of x ∈ E∗ inH1(E,Q/Z(1)), x ∪ [A] ∈ H3(E,Q/Z(2)) is the cup-product and NE/F is thenorm map.

Denote by A1, . . . , Am the separable algebras associated with X1, . . . , Xm

respectively. Then A = A1 × · · · × Am and E = E1 × · · · × Em. Thus for anyx ∈ E∗

NE/F (x ∪ [A]) = NE1/F (x1 ∪ [A1]) + · · ·+NEm/F (xm ∪ [Am]) ,

where xi is the Ei-component of x, what proves that ker f = 0.

Corollary 2.3. Let X1, . . . , Xm and X ′1, . . . , X

′m be homogeneous vari-

eties such that Xi is stably birationally equivalent to X ′i for i = 1, . . . ,m. The

quotientTorsCH2(X1 × · · · ×Xm)


is isomorphic to the quotient

TorsCH2(X ′1 × · · · ×X ′

m)

pr∗1 TorsCH2(X ′

1) + · · ·+ pr∗mTorsCH2(X ′m)

.

Lemma 2.4. For any homogeneous variety X of dimension ≤ 2, the groupCH2(X) is torsion-free.

Proof. Since X is a homogeneous variety, K(X) is a torsion-free group([65]). Since dimX ≤ 2, the term K(X)(3) of the topological filtration istrivial. Hence K(X)(2/3) is a torsion-free group. By [81, §9], CH2(X) ≃K(X)(2/3). Hence TorsCH2(X) = 0.

Corollary 2.5. Under the conditions of Corollary 2.3 suppose addition-ally that the varieties X1, . . . , Xm;X

′1, . . . , X

′m have the dimensions ≤ 2. Then

there is an isomorphism

TorsCH2(X1 × · · · ×Xm) ≃ TorsCH2(X ′1 × · · · ×X ′

m).

Proof. Obvious in view of Corollary 2.3 and Lemma 2.4.

Lemma 2.6. Let X1 and X2 be homogeneous varieties. If the variety(X2)F (X1) has a rational point, then

H3(F (X1 ×X2)/F,Q/Z(2)) = H3(F (X1)/F,Q/Z(2)) .

Proof. Since the homogeneous variety (X2)F (X1) has a rational point, itis rational, i.e. the field extension F (X1×X2)/F (X1) is purely transcendental.

Corollary 2.7. Let X1 and X2 be projective quadrics of the dimensions≤ 2. If the quadric (X2)F (X1) is isotropic (e.g., if X2 is isotropic or if X1 ≃ X2)

then TorsCH2(X1 ×X2) = 0.


Proof. Follows from Lemma 2.6, Proposition 2.2 and Lemma 2.4.

Lemma 2.8. For any quadratic form ρ of dimension ≥ 3, we have

2H3(F (ρ)/F,Q/Z(2)) = 0.

In other words, H3(F (ρ)/F,Q/Z(2)) = H3(F (ρ)/F ).

Proof. Let u ∈ H3(F (ρ)/F,Q/Z(2)). There exists a field extension L/Fsuch that ρL is isotropic and [L : F ] ≤ 2. Since ρL is isotropic, uL = 0. Usingtransfer homomorphism, we have [L : F ] · u = 0. Hence 2u = 0.

Corollary 2.9. For any quadratic form ρ of dimension ≥ 3 the ho-momorphism H3(F (ρ)/F ) → TorsCH2(Xρ), induced by the epimorphism ofProposition 2.1, is surjective. In particular, 2TorsCH2(Xρ) = 0.

Lemma 2.10. Let ρ1 and ρ2 be quadratic form of dimension ≥ 3. Then

2H3(F (ρ1, ρ2)/F,Q/Z(2)) = 0.

In other words, H3(F (ρ1, ρ2)/F,Q/Z(2)) = H3(F (ρ1, ρ2)/F ).

Proof. Let ρ′1 and ρ′2 be 3-dimensional subforms in ρ1 and ρ2 respec-tively. Clearly H3(F (ρ1, ρ2)/F,Q/Z(2)) ⊂ H3(F (ρ′1, ρ

′2)/F,Q/Z(2)). Thus,

replacing ρ1 by ρ′1 and ρ2 by ρ

′2, one can reduce the proof to the case dim ρ1 =

dim ρ2 = 3. In this case, dimXρ1×Xρ2 = 2; therefore TorsCH2(Xρ1×Xρ2) = 0(Lemma 2.4). For i = 1, 2, the conic Xρi is isomorphic to the Severi-Brauer

variety of the algebra Cidef= C0(ρi). Applying [66, Thm. 41], we obtain

H3(F (ρ1, ρ2)/F,Q/Z(2)) = [C1] ∪H1(F,Q/Z(1)) + [C2] ∪H1(F,Q/Z(1)) .Since 2[C1] = 2[C2] = 0 in the group H2(F,Q/Z(1)) = Br(F ), it follows that2H3(F (ρ1, ρ2)/F,Q/Z(2)) = 0.

Corollary 2.11. Let ρ1 and ρ2 be quadratic forms of dimension ≥ 3.Then the homomorphism

H3(F (ρ1, ρ2)/F ) → TorsCH2(Xρ1×Xρ2)

induced by the epimorphism of Proposition 2.1, is surjective. In particular,2TorsCH2(Xρ1×Xρ2) = 0.

Corollary 2.12. For any quadratic forms ρ1 and ρ2 of dimension ≥ 3,there is a natural isomorphism

H3(F (ρ1, ρ2)/F )

H3(F (ρ1)/F ) +H3(F (ρ2)/F )≃ TorsCH2(Xρ1 ×Xρ2)

pr∗1TorsCH2(Xρ1) + pr∗2TorsCH

2(Xρ2).

Proof. Follows from Proposition 2.2 and Lemmas 2.8 and 2.10.

Corollary 2.13. For any quadratic forms ρ1 and ρ2 with 3 ≤ dim ρi ≤ 4(i = 1, 2), there is a natural isomorphism

H3(F (ρ1, ρ2)/F )

H3(F (ρ1)/F ) +H3(F (ρ2)/F )≃ TorsCH2(Xρ1 ×Xρ2).

Proof. Follows from Corollary 2.12 and Lemma 2.4.

3. THE GROTHENDIECK GROUP OF A QUADRIC 65

3. The Grothendieck group of a quadric

In this §, ρ is an (n+2)-dimensional quadratic form over F (where n ≥ 1),V is the vector space of definition of ρ, P is the projective space of the vectorspace dual to V , and X = Xρ ⊂ P is the n-dimensional projective quadricdetermined by ρ.

We are mainly interested in the case when n = 2, i.e. when X is a surface.The even Clifford algebra C0(ρ) of the form ρ is denoted in this § by C.

Let U be Swan’s sheaf on X [82, §6]. It is an C ⊗F OX-module locally free asOX-module (note that the algebra C is canonically self-opposite; thus it is notnecessary to distinguish between left and right action of C).

We denote by h the class of a general hyperplane section of X, i.e. thepull-back of the class of a hyperplane with respect to the imbedding X ↪→ P.The subring of K(X) generated by h is denoted by H; it coincides with theimage of the pull-back homomorphism K(P) → K(X). Some further evidentassertions on H are collected in

Lemma 3.1. The abelian group H is freely generated by 1, h, . . . , hn. Thetopological filtration on K(X) induces on H the filtration by powers of h, i.e.for every 0 ≤ r ≤ n, the term H(r) is generated by all hj with r ≤ j ≤ n. Inparticular, the adjoint graded group G∗H is torsion-free.

In the case when X splits (i.e. when ρ is hyperbolic) and n = 2, we referas to a line class (resp., point class) to the class in K(X) of a line (resp., of aclosed rational point) lying on X.

Lemma 3.2 ([31]). Suppose that X splits and dimX = 2.

1. For any two different lines in X, their classes in K(X) coincide if andonly if the lines have no intersection. There are exactly two differentline classes in K(X).

2. The classes in K(X) of any two closed rational points of X coincide,i.e. there is only one point class in K(X).

3. Denote by l and l′ the different line classes and by p the point classin K(X). The abelian group K(X) is freely generated by the elements1, l, l′, p.

4. The second term K(X)(2) of the topological filtration on K(X) is gener-ated by p; the term K(X)(1) is generated by l, l′, p.

5. The multiplication in K(X) is determined by the formulas l2 = 0 = (l′)2

and l · l′ = p.6. h = l + l′ − p.

In the case when the quadric X is arbitrary (not necessary of dimension 2,not necessary split), we dispose of the following information on K(X):

Lemma 3.3. 1. The group K(X) is torsion-free and, for any field ex-tension E/F , the restriction homomorphism K(X) → K(XE) is injec-tive.


2. The class [U(n)] ∈ K(X) of the n times twisted Swan’s sheaf equals

2n + 2n−1h+ · · ·+ 2hn−1 + hn .

3. The homomorphism u : K(C) → K(X) given by the functor of takingtensor product U(n)⊗C (−) induces an epimorphism K(C) � K(X)/H.

4. If C is a skewfield, then K(X) = H.5. For any autoisometry ξ of the quadratic form ρ, the diagram

K(C)u−−−→ K(X)x x

K(C)u−−−→ K(X)

commutes, where the vertical maps are induced by the automorphisms ofC and of X given by ξ.

Proof. 1. Follows from [82, Theorem 9.1].2. See [32, Lemma 3.6].3. According to [82, Theorem 9.1], the functor U ⊗C (−) induces an epimor-phism K(C) → K(X)/H. Since for any r ∈ Z (and in particular for r = n)the twisting by r gives an automorphism of K(X)/H, the functor U(n)⊗C (−)induces an epimorphism as well.4. If C is a skewfield, then the image of this epimorphism is generated by[U(n)]. Since [U(n)] ∈ H by Item 2, it follows that K(X) = H.5. It is evident in view of the way the sheaf U is constructed (see [82, §6]).

Lemma 3.4 ([48]). The F -algebra C = C0(ρ) has the dimension 2n+1 =2dim ρ−1 over F . Its isomorphism class depends only on the similarity class ofρ. Moreover,

• if n is odd, then C is a central simple F -algebra;• if n is even, then C ≃ C0(ρ

′) ⊗F F (√d± ρ) where ρ′ is an arbitrary

1-codimensional subform of ρ.

In particular, if ρ is an even-dimensional form of trivial discriminant, thealgebra C is the direct product of two isomorphic central simple algebras; anyautomorphism of C should either interchange or stabilize the factors.

Lemma 3.5. Suppose that dim ρ is even and d± ρ is trivial. Let ξ be anautoisometry of the quadratic space (V, ρ) having the determinant −1. Thenthe automorphism of C induced by ξ interchanges the simple components of C.

Proof. Since d± ρ is trivial, there exists a basis v0, . . . , vn+1 of V suchthat

(v0 · · · vn+1)2 = 1 ∈ C .

Since ξ(v0) · · · ξ(vn+1) = (det ξ)·(v0 · · · vn+1) = −v0 · · · vn+1, the automorphismof C induced by ξ interchanges the elements

e = (1 + v0 · · · vn+1)/2 and e′ = (1− v0 · · · vn+1)/2 .

3. THE GROTHENDIECK GROUP OF A QUADRIC 67

Since e and e′ are orthogonal idempotents, they lie in different simple compo-nents of C. Therefore, the components of C are interchanged.

Comparing Lemma 3.2 with Lemma 3.3 in the situation of a split quadricsurface X, we get the following computation (note that here C is isomorphic toM2(F )×M2(F ) and thus there exist exactly two, up to isomorphisms, simpleC-modules; their classes are free generators of K(C)):

Lemma 3.6. Suppose that X splits and dimX = 2. There exist simpleC-modules M and M ′ such that u = 1 + l and u′ = 1 + l′ where

udef= u([M ]) = [U(2)⊗C M ], u′

def= u([M ′]) = [U(2)⊗C M

′] ∈ K(X) .

Proof. Take as M an arbitrary simple C-module and denote by M ′ a(determined uniquely up to an isomorphism) simple C-module non-isomorphicto M . Since by Lemma 3.2 the elements 1, l, l′, p generate K(X), we have

u = a+ bl + b′l′ + cp

for certain a, b, b′, c ∈ Z. Now we are going to show that

u′ = a+ b′l + bl′ + cp .

Let ξ be an autoisometry of the quadratic space (V, ρ) having determinant−1. By Lemma 3.5, the induced by ξ automorphism of K(C) interchanges[M ] and [M ′]. Thus, by Item 5 of Lemma 3.3, the induced by ξ automorphismof K(X) interchanges u and u′.

Since ρ splits, there exist 2-dimensional totally isotropic subspaces W andW ′ of V with 1-dimensional intersection and an autoisometry ξ of (V, ρ) hav-ing the determinant −1 interchanging W and W ′. The line classes in K(X)determined by W and W ′ are different (Item 1 of Lemma 3.2); therefore theycoincide with l and l′ (or vice versa: with l′ and l).

Thus, we have found an automorphism of K(X) interchanging u with u′

and l with l′ while leaving untouched 1 (of course) and p (since all the pointclasses coincide). Thereby, u′ = a+ b′l + bl′ + cp.

Since 2([M ] + [M ′]) = [C] ∈ K(C), we have: 2(u + u′) = [U(2)], and so,2(u + u′) = 4 + 2h + h2 by Item 2 of Lemma 3.3. Since K(X) is torsion-free,the last equality can be divided by 2. Replacing h by l + l′ − p and h2 by(l+ l′− p)2 = 2p (see Lemma 3.2), we obtain that u+u′ = 2+ l+ l′. From theother hand, u+ u′ = 2a+ (b+ b′)l + (b′ + b)l′ + 2c; therefore a = 1, b+ b′ = 1and c = 0.

We have proved that

u = 1 + bl + (1− b)l′ and u′ = 1 + (1− b)l + bl′

for certain b ∈ Z. It remains to show that b = 1 or b = 0.It follows from Item 3 of Lemma 3.3 that the elements u, u′, 1, h, h2 generate

the group K(X). Since h2 = 2p and h = u+ u′ − 2− p, the elements u, u′, 1, palso generate K(X). So, the quotient K(X)/(Z ·1+Z ·p) which is according to


Item 6 of Lemma 3.2 freely generated by l, l′ is also generated by u, u′. Thus,the Z-matrix (

b 1− b1− b b

)is invertible, i.e. its determinant is ±1. Hence, b = 1 or b = 0.

4. The Grothendieck group of a product of quadrics

In this and in the next §, we work with two quadratic forms ρ1 and ρ2 of thedimensions ≥ 3. We use the notation of the previous § amplified by the index1 or 2. So, for i = 1, 2, we have ρi, ni (we are mainly interested in the casewhen n1 = 2 = n2), Vi, Pi, Xi, Ci, Ui,hi, Hi, li, l

′i and pi. We set n = (n1, n2),

P = P1 × P2, X = X1 ×X2, and C = C1 ⊗F C2.For any x1 ∈ K(X1) and x2 ∈ K(X2), we denote by x1 � x2 the product

pr∗1(x1) · pr∗2(x2) ∈ K(X) where pr1 and pr2 are the projections of X1 ×X2 onX1 and X2 respectively.

Denote by H the image of the pull-back homomorphism K(P) → K(X).

Lemma 4.1. One has: H = H1 � H2 ⊂ K(X). The abelian group H isfreely generated by all hj11 �hj22 with 0 ≤ j1 ≤ n1 and 0 ≤ j2 ≤ n2. Moreover, thefiltration on H induced by the topological filtration on K(X) looks as follows:for any 0 ≤ r ≤ n1 + n2, the term H(r) is generated by all hj11 � hj22 withj1 + j2 ≥ r. In particular, the adjoint graded group G∗H is torsion-free.

The following lemma is also evident; together with Lemma 3.2, it gives acomplete description of the ring with filtration K(X) in the split situation.

Lemma 4.2. If X1 and X2 split then the map K(X1) ⊗K(X2) → K(X),x1 ⊗ x2 7→ x1 � x2 is an isomorphism of rings with filtrations.

For an OX1-module F1 and an OX2-module F2, we denote by F1 � F2 thetensor product pr∗1(F1)⊗OX pr

∗2(F2). The sheaf U = U1�U2 has for i = 1, 2 the

structures of a Ci-module commuting with each other. Thus, it is a C-module.Set U(n) = U1(n1)� U2(n2). It is also a C-module. The functor of taking thetensor product U(n)⊗C (−) determines a homomorphism u : K(C) → K(X).

Lemma 4.3. 1. The group K(X) is torsion-free and, for any field ex-tension E/F , the restriction homomorphism K(X) → K(XE) is injec-tive.

2. The homomorphism u : K(C) → K(X), defined right above, induces anepimorphism K(C) � K(X)/

(K(X1)�K(X2)

).

3. If C is a skewfield, then K(X) = H.

Proof. 1. This statement is valid for any homogeneous variety X ([65]).2. The isomorphism K∗(X1) ≃ K∗(F )

⊕n1 ⊕ K∗(C1) of [82, Theorem 9.1]remains bijective after changing the base F to any field extension, i.e. for anyfield extension E/F , the homomorphism K∗(SpecE×X1) → K∗(SpecE)

⊕n1⊕K∗(SpecE,C1) is bijective. Therefore, for any F -variety Y , the defined in the

5. CH2 OF A PRODUCT OF QUADRICS 69

similar way homomorphism K∗(Y ×X1) → K∗(Y )⊕n1 ⊕K∗(Y,C1) is bijective(compare to the proof of Proposition 4.1 of [69, §7]). In particular, K(X) ≃K(X2)

⊕n1⊕K(X2, C1) . ComputingK(X2) andK(X2, C1) using [82, Theorem9.1] once again, one gets

K(X) ≃ K(F )⊕n1n2 ⊕K(C1)⊕n2 ⊕K(C2)

⊕n1 ⊕K(C) .

The image of K(F )⊕n1n2 ⊕ K(C1)⊕n2 ⊕ K(C2)

⊕n1 in K(X) is contained inK(X1) � K(X2) and the homomorphism K(C) → K(X) is induced by thefunctor of taking tensor product U ⊗C (−). Thus u : K(C) → K(X) is moduloK(X1)�K(X2) an epimorphism.3. If the algebra C is a skewfield then the image of u is contained in H;moreover, the algebras C1 and C2 are skewfields as well and thus K(Xi) = Hi

for i = 1, 2.

Corollary 4.4. If C is a skewfield, then G∗K(X) is torsion-free. Inparticular, TorsCH2(X) = 0.

Proof. If C is a skewfield, then K(X) = H by Item 3 of Lemma 4.3.Consequently, TorsG∗K(X) = TorsG∗H = 0 (see Lemma 4.1).

5. CH2 of a product of quadrics

The notation used in this § is introduced in the beginning of the previousone. However, each of the quadratic forms ρ1 and ρ2 is now supposed to havethe dimension 3 or 4. So, each of Xi is either a quadric surface or a conic. Weare mainly interested in the case when X1 and X2 are surfaces.

Theorem 5.1. Suppose that dim ρ1 = 4 = dim ρ2, i.e. that X1 and X2

are surfaces. If det ρ1 = det ρ2, then TorsCH2(X1 ×X2) = 0.

Proof. If one of the quadratic forms is isotropic, then TorsCH2(X1 ×X2) = 0 by Corollary 2.7. In the rest of the proof we assume that ρ1 and ρ2are anisotropic.

As a next step, we are going to consider the case when det ρ1 = det ρ2 = 1.

Lemma 5.2. Any projective quadric surface defined by a quadratic form ofdeterminant 1 is stably birationally equivalent to a conic.

Proof. Suppose that we are given a quadric determined by a 4-dimensionalquadratic form ρ with det ρ = 1. Take the conic determined by an arbitrary3-dimensional subform ρ′ ⊂ ρ. Since ρ′ becomes isotropic over F (ρ) and viceversa, ρ becomes isotropic over F (ρ′), the quadrics given by ρ′ and ρ are stablybirationally equivalent.

Suppose that det ρ1 = det ρ2 = 1 and choose some conics X ′1 and X

′2 stably

birationally equivalent to X1 and X2 respectively. Applying Corollary 2.5, weget an isomorphism of TorsCH2(X1 ×X2) onto the group TorsCH2(X ′

1 ×X ′2)

which is trivial by Lemma 2.4.Therefore, we may assume that d = 1 where d = det ρ1 = det ρ2.


As a next step of the proof of Theorem, we consider the case when the

F -algebras C1def= C0(ρ1) and C2

def= C0(ρ2) are isomorphic. In this case, the

forms ρ1 and ρ2 becomes similar over the field F (√d). Thus by a theorem

of Wadsworth ([86, Theorem 7]), they are already similar over F . Thereforethe quadrics X1 and X2 are isomorphic and consequently TorsCH2(X) = 0 byCorollary 2.7.

It remains only to consider the situation when the forms ρ1 and ρ2 areanisotropic, d = 1 and C1 ≃ C2. Set c = indC. We have: c = 2 or c = 4.

Fix a separable closure F of the field F . For the algebra CF , the varietyXF , etc. we shall use the notation C, X, etc.

For i = 1, 2, denote by Mi and M ′i the (determined uniquely up to an

isomorphism and up to the order) non-isomorphic simple Ci-modules. Thereare exactly 4 different isomorphism classes of simple C-modules; they are rep-resented by M1 �M2 (M1 �M2 is by definition the tensor product M1 ⊗M2

considered as C-module in the natural way),M1�M ′2,M

′1�M2, andM

′1�M ′

2.Denote by mi the class of Mi and by m′

i the class of M′i in K(Ci). The abelian

group K(C) is freely generated by m1 � m2 (m1 � m2 is defined as follows:for i = 1, 2, one takes the image of mi ∈ K(Ci) with respect to the mapK(Ci) → K(C) and than takes the product of the images in the ring K(C)),m1�m′

2, m′1�m2, and m

′1�m′

2. We identify K(C) with a subgroup in K(C)via the restriction map K(C) ↪→ K(C).

Lemma 5.3. The subgroup K(C) ⊂ K(C) is generated by

c · (m1 �m2 +m′1 �m′

2) and c · (m1 �m′2 +m′

1 �m2) .

Proof. Denote by L the quadratic extension F (√d) of the field F , where

d = det ρ1 = det ρ2. The algebra CL is the direct product of 4 copies of acentral simple L-algebra of index c. Evidently, the subgroup K(CL) of K(C)is freely generated by c ·m1 �m2, c ·m1 �m′

2, c ·m′1 �m2, and c ·m′

1 �m′2.

Now we are going to determine K(C) as a subgroup in K(CL). Computingthe norm NL/F : K(CL) → K(C), we get:

xdef= NL/F (c ·m1 �m2) = c · (m1 �m2 +m′

1 �m′2) ;

x′def= NL/F (c ·m1 �m′

2) = c · (m1 �m′2 +m′

1 �m2) .

Thus, the elements x and x′ are in K(C). Note that:

• x and x′ can be included in a system of free generators of the free abeliangroup K(CL) (e.g. x, x

′, c ·m1 �m2, and c ·m1 �m′2);

• K(C) is a free abelian group of rank 2 (because the algebra C is thedirect product of two copies of a simple algebra, since for i = 1, 2 onehas: Ci = C ′

i ⊗F L for a central simple F -algebra C ′i);

• K(C) is a subgroup of K(CL) containing x and x′.

Consequently, K(C) is generated by x and x′.


We identify K(X) with a subgroup in K(X) via the restriction map (whichis injective by Item 1 of Lemma 4.3). For i = 1, 2, let li, l

′i be the different line

classes and pi the point class in K(Xi) (see Lemma 3.2).

Corollary 5.4. The group K(X) is generated modulo H by c · (l1 � l2 +l′1 � l′2) and c · p1 � p2.

Proof. According to Item 2 of Lemma 4.3, the map u : K(C) → K(X)/His surjective. By Lemma 5.3, the group K(C) is generated by

c · (m1 �m2 +m′1 �m′

2) and c · (m1 �m′2 +m′

1 �m2) .

Applying Lemma 3.6, we can compute the images of these generators in K(X):up to the order, they are

c ·((1 + l1)� (1 + l2) + (1 + l′1)� (1 + l′2)

)and

c ·((1 + l1)� (1 + l′2) + (1 + l′1)� (1 + l2)

).

One can modify the first expression as follows (the formulas of Lemma 3.2 arein use):

c ·((1 + l1)� (1 + l2) + (1 + l′1)� (1 + l′2)

)=

= c ·(2 + (l1 + l′1)� 1 + 1� (l2 + l′2) + l1 � l2 + l′1 � l′2

)=

= c ·(2 + (h1 + h21/2)� 1 + 1� (h2 + h22/2) + l1 � l2 + l′1 � l′2

)≡

≡ c · (l1 � l2 + l′1 � l′2) (mod H)

(note that c is divisible by 2). The analogous modification can be made for thesecond expression as well. Thus, the group K(X) is generated modulo H byc · (l1� l2+ l′1� l′2) and c · (l1� l′2+ l′1� l2). Taking the sum of these generators,we get:

c · (l1 � l2 + l′1 � l′2) + c · (l1 � l′2 + l′1 � l2) =

= c · (l1 + l′1)� (l2 + l′2) = c · (h1 + h21/2)� (h2 + h22/2) ≡≡ c · (h21/2)� (h22/2) = c · p1 � p2

(where the congruence is modulo H).

Lemma 5.5. 1. c · (l1 � l2 + l′1 � l′2) ∈ K(X)(2);2. c · p1 � p2 ∈ K(X)(3);3. for any 0 = r ∈ Z, the set r · c(l1 � l2 + l′1 � l′2) +H has no intersection

with K(X)(3).

Proof. 1. It is evident that c(l1� l2+ l′1� l′2) ∈ K(X)(2). Since K(X)(2) =K(X)(2) ∩K(X) (see e.g. [74, Lemme 6.3, (i)]), we are done.


2. If we multiply the element c(l1 � l2 + l′1 � l′2) ∈ K(X)(2) by the elementh1 � 1 ∈ K(X)(1), we get:

K(X)(3) ∋ c(l1 � l2 + l′1 � l′2) · (h1 � 1) =

= c(p1 � l2 + p1 � l′2) = c · p1 � (h2 + p2) =

= c · p1 � h2 + c · p1 � p2 .

Since c · p1 � h2 ∈ H(3) ∈ K(X)(3), it follows that c · p1 � p2 ∈ K(X)(3).3. By Lemmas 3.2 and 4.2, the abelian group K(X) is freely generated by theproducts x1 � x2 where xi is one of the elements 1, li, l

′i, pi; moreover, the term

K(X)(3) of the filtration is generated by l1 � p2, l′1 � p2, p1 � l2, p1 � l′2 and

p1 � p2. In particular, 4K(X)(3) ⊂ H.Suppose that, for certain 0 = r ∈ Z, the intersection of r·c(l1�l2+l′1�l′2)+H

withK(X)(3) is non-empty. Then 4r·c(l1�l2+l′1�l′2) ∈ H, a contradiction.

Corollary 5.6. Let us supply the quotient K(X)/H with the filtrationinduced from K(X). Then TorsG2(K(X)/H) = 0.

Proof. By Corollary 5.4 and Lemma 5.5, G2(K(X)/H) is an infinitecyclic group (generated by the residue of c(l1 � l2 + l′1 � l′2)).

To finish the proof of Theorem 5.1, consider the exact sequence

0 → G2H → G2K(X) → G2(K(X)/H) → 0 .

The left-hand side term is torsion-free by Lemma 4.1 while the right-handside term is torsion-free by Corollary 5.6. Consequently, the middle term is atorsion-free group as well.

Theorem 5.7. The order of the group TorsCH2(X1 ×X2) is at most 2.

Proof. Since 2TorsCH2(X1 × X2) = 0 by Corollary 2.11, it suffices toshow that the torsion in CH2(X1 ×X2) is a cyclic group.

By Corollary 2.7, it suffices to consider only the case when the both qua-dratic forms ρ1 and ρ2 are anisotropic.

Set as usual X = X1 ×X2, Ci = C0(ρi) and C = C1 ⊗F C2. Suppose thatthe algebra C is simple. Then K(C) is a cyclic group and therefore, by Item2 of Lemma 4.3, the quotient K(X)/H is cyclic as well. Moreover, C1 andC2 are division algebras (since they are simple and the quadratic forms areanisotropic) and therefore K(Xi) = Hi for i = 1, 2 by Item 4 of Lemma 3.3.Supplying K(X)/H with the filtration induced from K(X), we get an exactsequence of the adjoint graded groups

0 → G∗H → G∗K(X) → G∗(K(X)/H) → 0 .

Take any r ≥ 0. Since GrH is torsion-free (Lemma 4.1), TorsGrK(X) ismapped injectively into Gr(K(X)/H). Since K(X)/H is cyclic, Gr(K(X)/H)is cyclic as well and thus so is also TorsGrK(X). In particular, the groupTorsCH2(X) ≃ TorsG2K(X) is cyclic.

Now suppose that C is not simple. Then


either: dimX1 = 2 = dimX2 and detX1 = detX2,or: for i = 1 or for i = 2, one has: dimXi = 2 and detXi = 1.

In the first case, the torsion in CH2(X) is 0 by Theorem 5.1. In the secondcase, we replace the surface Xi by a stably birationally equivalent conic (seeLemma 5.2 and Corollary 2.5).

Theorem 5.8. If indC0(ρ1)⊗F C0(ρ2) = 4, then TorsCH2(X1×X2) = 0.

Proof. We set C = C0(ρ1)⊗F C0(ρ2) and suppose that indC = 4.If C is a simple algebra, then it is a skewfield and we are done by Corollary

4.4.If C is not simple, then

either: dimX1 = 2 = dimX2 and det ρ1 = det ρ2,or: for i = 1 or for i = 2, one has: dimXi = 2 and detXi = 1.

In the first case, the torsion in CH2(X1 × X2) is 0 by Theorem 5.1. In thesecond case, we replace the surface Xi by a stably birationally equivalent conic(see Lemma 5.2 and Corollary 2.5).

Theorem 5.9. Suppose that dim ρ1 = 4, det ρ1 = 1 and that for a certain3-dimensional subform ρ′1 of ρ1 one has:

indC0(ρ1)⊗F C0(ρ2) = indC0(ρ′1)⊗F C0(ρ2) .

Then TorsCH2(X1 ×X2) = 0.

Proof. Applying the same arguments as above, we may assume that

• the forms ρ1 and ρ2 are anisotropic and• one of the following alternative conditions holds:

– the dimension of ρ2 equals 3 or– the dimension of ρ2 is 4 and det ρ1 = det ρ2 = 1.

We are going to show that, under the assumptions made, TorsG2K(X1×X2) =0.

The algebra C is now simple; it has the index 1, 2, or 4. Set c = indC.The group K(C) is generated by (c/4) · [C] where [C] ∈ K(C) is the class ofC.

Consider the case when dim ρ2 = 4.It follows from Item 2 of Lemma 4.3 that K(X) is generated modulo H by

the element (c/4)[U(2, 2)]. Applying Item 2 of Lemma 3.3, one computes that[U(2, 2)] = (4 + 2h1 + h21)� (4 + 2h2 + h22) ∈ K(X). Thus, K(X) is generated

modulo H also by xdef= (c/4)(2 · h1 � h22 +2 · h21 � h2 + h21 � h22). Since we have

the exact sequence

0 → G∗H → G∗K(X) → G∗(K(X)/H) → 0

with torsion-free G∗H, it would suffice to show that x ∈ K(X)(3).Consider the conic X ′

1 determined by ρ′1 and denote by U ′1 Swan’s sheaf on

X ′1. The product U ′

1(1) � U2(2) of twisted Swan’s sheaves has a structure of

module over C ′ def= C ′

1 ⊗ C2; its class in K(X ′), where X ′ def= X ′

1 ×X2 is equal


to (2 + h′1) � (4 + 2h2 + h22) where h′1 is the class in K(X ′1) of a hyperplane

section of X ′1. Since indC ′ = indC = c, the latter product can be divided by

(4/c) in K(X ′), i.e.

K(X ′) ∋ x′def= (c/4)(2 · 1� h22 + 2 · h′1 � h2 + h′1 � h22) .

Since 4x′ ∈ K(X ′)(2) and the group G1K(X ′) = CH1(X ′) is torsion-free (seee.g. [74, Lemme 6.3, (i)]), it follows that x′ ∈ K(X ′)(2). Since the image ofx′ with respect to the push-forward given by the closed imbedding X ′ ↪→ Xcoincides with x and codimX X

′ = 1, the element x is in K(X)(3).Now suppose that dim ρ2 = 3.If c = 1, then the quadric (X2)F (X1) is isotropic and therefore TorsCH2(X) =

0 by Corollary 2.7. Thus we may assume that c is divisible by 2.The group K(X) is now generated modulo H by (c/4)[U(2, 1)] and

[U(2, 1)] = (4 + 2h1 + h21)� (2 + h2) ∈ K(X) .

Thus, K(X) is generated modulo H also by xdef= (c/4)(h21 � h2) and it suffices

to show that x ∈ K(X)(3).The class in K(X ′) of the product U ′

1(1)�U2(1) of twisted Swan’s sheavesis equal this time to (2+ h′1)� (2+ h2) and can be divided by (4/c) in K(X ′),i.e.

K(X ′) ∋ x′def= (c/4)(h′1 � h2) .

Since x′ ∈ K(X ′)(2) and the image of x′ with respect to the push-forwardgiven by the closed imbedding X ′ ↪→ X coincides with x, the element x is inK(X)(3).

Corollary 5.10. If ρ1 and ρ2 contain similar 3-dimensional subforms,then TorsCH2(X1 ×X2) = 0.

Proof. If dim ρ1 = 3 or if det ρ1 = 1, then the quadric (X2)F (X1) isisotropic and so we are done by Corollary 2.7.

Therefore, we may assume that dim ρ1 = 4 and det ρ1 = 1. These are thefirst two conditions of Theorem 5.9. We state that also the last condition ofTheorem 5.9 is satisfied. Indeed, denote by ρ′1 ⊂ ρ1 and ρ′2 ⊂ ρ2 the similar3-dimensional subforms. According to Lemma 3.4, the F -algebras C0(ρ

′1) and

C0(ρ′2) are isomorphic and C0(ρi) = C0(ρ

′i)F (

√det ρi) for i = 1, 2. Therefore,

indC0(ρ1)⊗F C0(ρ2) = 1 = indC0(ρ′1)⊗F C0(ρ2) .

6. The group I3(F (ρ, ψ)/F )

The following assertion is obvious:

Lemma 6.1. Let ρ = ⟨−a,−b, ab, d⟩ be a quadratic form over F . For anyk ∈ F ∗ the following conditions are equivalent.

(1) k ∈ DF (⟨⟨d⟩⟩);(2) ⟨⟨a, b, k⟩⟩ = ρ ⟨⟨k⟩⟩;(3) ρ ⟨⟨k⟩⟩ ∈ P3(F ).

7. THE CASE OF INDEX 1 75

Lemma 6.2. Let ρ = ⟨−a,−b, ab, d⟩ be a quadratic form over F . Then

1. P3(F (ρ)/F ) = {⟨⟨a, b, k⟩⟩ | k ∈ DF (⟨⟨d⟩⟩)},2. H3(F (ρ)/F ) = {(a, b, k) | k ∈ DF (⟨⟨d⟩⟩)}.Proof. 1. See [15, Lemma 3.1].

2. Let ρ0 = ⟨−a,−b, ab⟩. Clearly H3(F (ρ)/F ) ⊂ H3(F (ρ0)/F ). It followsfrom [5, Beweis vom Satz 5.6] that H3(F (ρ0)/F ) = (a, b)∪H1(F ). Hence anyelement u ∈ H3(F (ρ)/F ) has the form (a, b, x) where x ∈ F ∗. Since (a, b, x) ∈H3(F (ρ)/F ), the Pfister form ⟨⟨a, b, x⟩⟩F (ρ) is hyperbolic. It follows from the

first assertion that there exists k ∈ DF (⟨⟨d⟩⟩) such that ⟨⟨a, b, x⟩⟩ = ⟨⟨a, b, k⟩⟩.Hence u = (a, b, x) = (a, b, k).

Corollary 6.3. Let ρ1, . . . , ρm be 4-dimensional quadratic forms over F .Then for a quadratic form ϕ the following conditions are equivalent:

(1) ϕ ∈ I3(F (ρ1)/F ) + · · ·+ I3(F (ρm)/F ) + I4(F );(2) ϕ ∈ P3(F (ρ1)/F ) + · · ·+ P3(F (ρm)/F ) + I4(F );(3) ϕ ∈ I3(F ) and e3(ϕ) ∈ H3(F (ρ1)/F ) + · · ·+H3(F (ρm)/F ).

Proof. (2)⇒(1)⇒(3). Obvious.(3)⇒(2). Follows from Lemma 6.2.

Corollary 6.4. Let ρ1, . . . , ρm be 4-dimensional quadratic forms suchthat H3(F (ρ1, . . . , ρm)/F ) = H3(F (ρ1)/F ) + · · ·+H3(F (ρm)/F ). Then

I3(F (ρ1, . . . , ρm)/F ) ⊂ I3(F (ρ1)/F ) + · · ·+ I3(F (ρm)/F ) + I4(F ).

Corollary 6.5. Let ρ = ⟨−a,−b, ab, d⟩ and ψ = ⟨−u,−v, uv, δ⟩ be qua-dratic forms over F . Then for any π ∈ I3(F (ρ)/F ) + I3(F (ψ)/F ) + I4(F )there exist k1, k2 ∈ F ∗ with the following properties:

1) ⟨⟨a, b, k1⟩⟩ = ρ ⟨⟨k1⟩⟩ and ⟨⟨u, v, k2⟩⟩ = ψ ⟨⟨k2⟩⟩;2) π ≡ ⟨⟨a, b, k1⟩⟩+ ⟨⟨u, v, k2⟩⟩ (mod I4(F )).

Proof. By Corollary 6.3, we have π ∈ P3(F (ρ)/F )+P3(F (ψ)/F )+I4(F ).

Hence there exist π1 ∈ P3(F (ρ)/F ) and π2 ∈ P3(F (ψ)/F ) such that

π ≡ π1 + π2 (mod I4(F )) .

By Lemma 6.2, there exist k1, k2 ∈ F ∗ such that π1 = ⟨⟨a, b, k1⟩⟩ and π2 =⟨⟨u, v, k2⟩⟩. Finally, Lemma 6.1 shows that ⟨⟨a, b, k1⟩⟩ = ρ ⟨⟨k1⟩⟩, ⟨⟨u, v, k2⟩⟩ =ψ ⟨⟨k2⟩⟩.

7. The case of index 1

In this §, we study the group H3(F (ρ, ψ)/F ) in the case where ρ, ψ are4-dimensional quadratic forms with non-trivial discriminants and indC0(ρ)⊗F

C0(ψ) = 1. In the case d± ρ = d± ψ we obviously have C0(ρ) ≃ C0(ψ). Henceρ is similar to ψ (see [86, Theorem 7]) and hence the group H3(F (ρ, ψ)/F )coincides with H3(F (ρ)/F ). So it is sufficient to study only the case where

d± ρ = d± ψ.


Replacing ρ and ψ by similar forms, we can rewrite our conditions as fol-lows:

1) ρ = ⟨−a,−b, ab, d⟩ and ψ = ⟨−u,−v, uv, δ⟩ with a, b, d, u, v, δ ∈ F ∗;2) d, δ, and dδ are not squares in F ∗;3) ind((a, b)⊗F (u, v))F (

√d,√δ) = 1.

During this section we will suppose that the conditions 1)–3) hold.We define the set Γ(ρ, ψ) as

{γ ∈ I3(F ) | there exist l1, l2 ∈ F ∗ such that γ = l1ρ+ l2ψ + ⟨⟨dδ⟩⟩}.Lemma 7.1. The set Γ(ρ, ψ) is not empty.

Proof. Since ind((a, b) ⊗F (u, v))F (√d,√δ) = 1, there exist s, r ∈ F ∗ such

that (a, b) ⊗ (u, v) = (d, s) ⊗ (δ, r). Set l1 = δs, l2 = −δr. It is sufficient to

verify that γdef= l1ρ+ l2ψ + ⟨⟨dδ⟩⟩ ∈ I3(F ). We have

γ = δsρ− δrψ + ⟨1,−dδ⟩ = δ(sρ− rψ + ⟨δ,−d⟩) == δ(s(⟨⟨a, b⟩⟩ − ⟨⟨d⟩⟩)− r(⟨⟨u, v⟩⟩ − ⟨⟨δ⟩⟩) + (⟨⟨d⟩⟩ − ⟨⟨δ⟩⟩)) == δ(s ⟨⟨a, b⟩⟩ − r ⟨⟨u, v⟩⟩+ ⟨⟨d, s⟩⟩ − ⟨⟨δ, r⟩⟩) .

Therefore γ ∈ I2(F ) and c(γ) = (a, b) + (u, v) + (d, s) + (δ, r) = 0. Henceγ ∈ I3(F ).

Lemma 7.2. Γ(ρ, ψ) ⊂ I3(F (ρ, ψ)/F ).

Proof. Let γ = l1ρ + l2ψ + ⟨⟨dδ⟩⟩ ∈ Γ(ρ, ψ). We have dim(γF (ψ,ρ))an ≤dim(ρF (ρ))an+dim(ψF (ψ))an+dim ⟨⟨δd⟩⟩ ≤ 2+2+2 = 6 < 8. Since γ ∈ I3(F ),the Arason-Pfister Hauptsatz shows that γF (ψ,ρ) is hyperbolic.

Corollary 7.3. For any γ ∈ Γ(ρ, ψ), we have e3(γ) ∈ H3(F (ρ, ψ)/F ).

Lemma 7.4. Let l, k ∈ F ∗ and let τ be a quadratic form such that τ ⟨⟨k⟩⟩ ∈I3(F ). Then lτ − ⟨⟨k⟩⟩ τ ≡ lkτ (mod I4(F )).

Proof. lτ − ⟨⟨k⟩⟩ τ − lkτ = −⟨⟨l⟩⟩ ⟨⟨k⟩⟩ τ ∈ ⟨⟨l⟩⟩ I3(F ) ⊂ I4(F ).

Lemma 7.5. Let γ ∈ Γ(ρ, ψ), π1 ∈ P3(F (ρ)/F ) and π2 ∈ P3(F (ψ)/F ).Then there exists γ′ ∈ Γ(ρ, ψ) such that γ − π1 − π2 ≡ γ′ (mod I4(F )). More-over, γ + π1 + π2 ≡ γ′ (mod I4(F )).

Proof. Let l1, l2 ∈ F ∗ be such that γ = l1ρ + l2ψ + ⟨⟨dδ⟩⟩. By Lemmas6.1 and 6.2, there exist k1, k2 ∈ F ∗ such that π1 = ρ ⟨⟨k1⟩⟩, π2 = ψ ⟨⟨k2⟩⟩. ByLemma 7.4, we have

l1ρ− π1 = l1ρ− ⟨⟨k1⟩⟩ ρ ≡ l1k1ρ (mod I4(F )),

l2ψ − π2 = l2ψ − ⟨⟨k2⟩⟩ψ ≡ l2k2ψ (mod I4(F )).

Hence γ − π1 − π2 ≡ l1k1ρ+ l2k2ψ + ⟨⟨dδ⟩⟩ (mod I4(F )). Setting γ′ = l1k1ρ+l2k2ψ + ⟨⟨dδ⟩⟩, we get the required equation γ − π1 − π2 ≡ γ′ (mod I4(F )).The second equation γ + π1 + π2 ≡ γ′ (mod I4(F )) is obvious in view of thecongruence πi ≡ −πi (mod I4(F )) (for i = 1, 2).


Corollary 7.6. Γ(ρ, ψ)+I3(F (ρ)/F )+I3(F (ψ)/F )+I4(F ) = Γ(ρ, ψ)+I4(F ).

Proof. It is an obvious consequence of Corollary 6.3 and Lemma 7.5

Lemma 7.7. The following conditions are equivalent:

(1) I3(F (ρ, ψ)/F ) ⊂ I3(F (ρ)/F ) + I3(F (ψ)/F ) + I4(F );(2) Γ(ρ, ψ) ⊂ I3(F (ρ)/F ) + I3(F (ψ)/F ) + I4(F );(3) there exists γ ∈ Γ(ρ, ψ) such that γ ∈ I3(F (ρ)/F )+I3(F (ψ)/F )+I4(F );(4) Γ(ρ, ψ) contains a hyperbolic form, i.e. 0 ∈ Γ(ρ, ψ);(5) the quadratic forms ψ and ρ contain similar 3-dimensional subforms;(6) TorsCH2(Xρ ×Xψ) = 0;(7) H3(F (ρ, ψ)/F ) = H3(F (ρ)/F ) +H3(F (ψ)/F ).

Proof. (1)⇒(2). Obvious in view of Lemma 7.2.(2)⇒(3). Obvious in view of Lemma 7.1.(3)⇒(4). Let γ be such as in (3). By Corollary 6.3, there exist π1 ∈ P3(F (ρ)/F )and π2 ∈ P3(F (ψ)/F ) such that γ ∈ π1+π2+I

4(F ). Hence γ−π1−π2 ∈ I4(F ).By Lemma 7.5, there exists γ′ ∈ Γ(ρ, ψ) such that γ−π1−π2 ≡ γ′ (mod I4(F )).Since γ − π1 − π2 ∈ I4(F ), we have γ′ ∈ I4(F ). By definition of Γ(ρ, ψ),dim(γ′)an ≤ 4 + 4 + 2 = 10 < 16. Since γ′ ∈ I4(F ), the Arason-PfisterHauptsatz shows that γ′ = 0.(4)⇒(5). Since 0 ∈ Γ(ρ, ψ), there exist l1, l2 ∈ F ∗ such that 0 = l1ρ + l2ψ +⟨⟨dδ⟩⟩. Thus l1ρ+ l2ψ = −⟨⟨dδ⟩⟩. Hence l1ρ and l2ψ contain a common subformof the dimension (dim(ρ) + dim(ψ)− dim ⟨⟨dδ⟩⟩)/2 = (4 + 4− 2)/2 = 3.(5)⇒(6). See Corollary 5.10.(6)⇒(7). See Corollary 2.13.(7)⇒(1). It is a particular case of Corollary 6.4.

Proposition 7.8. For an arbitrary element γ ∈ Γ(ρ, ψ), one has

H3(F (ρ, ψ)/F ) = H3(F (ρ)/F ) +H3(F (ψ)/F ) + e3(γ)H0(F ) .

Proof. By Corollary 7.3, the element e3(γ) belongs to H3(F (ρ, ψ)/F ).If TorsCH2(Xρ ×Xψ) = 0 then by Corollary 2.13, we have H3(F (ρ, ψ)/F ) =H3(F (ρ)/F )+H3(F (ψ)/F ) and the proof is complete. If TorsCH2(Xρ×Xψ) =0, Lemma 7.7 shows that γ /∈ I3(F (ρ)/F ) + I3(F (ψ)/F ) + I4(F ). Hence, byCorollary 6.3, e3(γ) /∈ H3(F (ρ)/F ) +H3(F (ψ)/F ). To complete the proof itis sufficient to apply Corollary 2.13 and Theorem 5.7.

Corollary 7.9.

I3(F (ρ, ψ)/F ) ⊂ I3(F (ρ)/F ) + I3(F (ψ)/F ) + {Γ(ρ, ψ), 0}+ I4(F ) .

Proof. Let τ ∈ I3(F (ρ, ψ)/F ). Choose an element γ ∈ Γ(ρ, ψ). ByProposition 7.8, either e3(τ) ∈ H3(F (ρ)/F ) + H3(F (ψ)/F ) or e3(τ − γ) ∈H3(F (ρ)/F ) +H3(F (ψ)/F ). It remains to apply Corollary 6.3.

Proposition 7.10. Let π ∈ I3(F (ρ, ψ)/F ). Then at least one of the fol-lowing conditions holds


1) π ∈ I3(F (ρ)/F ) + I3(F (ψ)/F ) + I4(F );2) π ∈ Γ(ρ, ψ) + I4(F ).

Proof. Obvious in view of Corollaries 7.9 and 7.6.

8. Main theorem

Proposition 8.1. Let ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩ be an anisotropic quadraticform. Let ψ = ⟨−u,−v, uv, δ⟩ and ρ = ⟨−a,−b, ab, d⟩. Then:

1. The following two conditions are equivalent:(i) ⟨⟨a, b, c⟩⟩ ∈ I3(F (ρ, ψ)/F ),(ii) ϕF (ψ) is isotropic.

2. The following two conditions are equivalent:(i) ⟨⟨a, b, c⟩⟩ ∈ I3(F (ρ)/F ) + I3(F (ψ)/F ) + I4(F ),(ii) there exits a 5-dimensional Pfister neighbor ϕ0 such that ϕ0 ⊂ ϕ

and (ϕ0)F (ψ) is isotropic.

Proof. Note that ⟨⟨a, b, c⟩⟩ = ϕ− cρ = ρ− cϕ.(1i)⇒(1ii). Let E = F (ψ). If the Pfister form ⟨⟨a, b, c⟩⟩E is isotropic, itsneighbor (⟨⟨a, b⟩⟩ ⊥ ⟨−c⟩)E is isotropic too. Since ⟨⟨a, b⟩⟩ ⊥ ⟨−c⟩ ⊂ ϕ, the formϕE is isotropic. Thus we can suppose that ⟨⟨a, b, c⟩⟩E is anisotropic. By theassumption, ⟨⟨a, b, c⟩⟩ ∈ I3(F (ρ, ψ)/F ) = I3(E(ρ)/F ). Hence the anisotropicPfister form ⟨⟨a, b, c⟩⟩E becomes isotropic over the function field of ρE. Bythe Arason-Pfister subform theorem, we have kρE ⊂ ⟨⟨a, b, c⟩⟩E where k is anarbitrary element of DE(ρ) ·DE(⟨⟨a, b, c⟩⟩). Since (ab)−1 ∈ DE(ρ) and −abc ∈DE(⟨⟨a, b, c⟩⟩) we can take k = (ab)−1 · (−abc) = −c. Thus −cρE ⊂ ⟨⟨a, b, c⟩⟩E.Hence dim((⟨⟨a, b, c⟩⟩ ⊥ cρ)E)an ≤ 8−4 = 4. Since ⟨⟨a, b, c⟩⟩+cρ = ϕ, it followsthat dim(ϕE)an ≤ 4. Hence ϕF (ψ) = ϕE is isotropic.(1ii)⇒(1i). Since ϕF (ψ) and ρF (ρ) are isotropic, we have dim(ϕF (ψ))an ≤ 4 anddim(ρF (ρ))an ≤ 2. Therefore dim(⟨⟨a, b, c⟩⟩F (ρ,ψ))an = dim((ϕ − cρ)F (ρ,ψ))an ≤4 + 2 = 6. By the Arason-Pfister theorem, ⟨⟨a, b, c⟩⟩F (ρ,ψ) is hyperbolic. Hence

⟨⟨a, b, c⟩⟩ ∈ I3(F (ρ, ψ)/F ).(2i)⇒(2ii). By Corollary 6.5, there exist k1, k2 ∈ F ∗ such that ⟨⟨a, b, k1⟩⟩ =ρ ⟨⟨k1⟩⟩, ⟨⟨u, v, k2⟩⟩ = ψ ⟨⟨k2⟩⟩, and

⟨⟨a, b, c⟩⟩ ≡ ⟨⟨a, b, k1⟩⟩+ ⟨⟨u, v, k2⟩⟩ (mod I4(F )) .

It follows from [9, Theorem 4.8] that the Pfister forms ⟨⟨a, b, c⟩⟩, ⟨⟨a, b, k1⟩⟩,and ⟨⟨u, v, k2⟩⟩ are linked. Hence there exists s ∈ F ∗ such that s ⟨⟨u, v, k2⟩⟩ =⟨⟨a, b, k1⟩⟩− ⟨⟨a, b, c⟩⟩. Since ⟨⟨a, b, k1⟩⟩ = ρ ⟨⟨k1⟩⟩ and ⟨⟨a, b, c⟩⟩ = ρ− cϕ, we haves ⟨⟨u, v, k2⟩⟩ = ρ ⟨⟨k1⟩⟩− (ρ− cϕ) = cϕ−k1ρ. Therefore ϕ− cs ⟨⟨u, v, k2⟩⟩ = ck1ρ.Hence ϕ and cs ⟨⟨u, v, k2⟩⟩ contain a common subform of the dimension

1

2(dimϕ+ dim(sc ⟨⟨u, v, k2⟩⟩)− dim(ck1ρ)) =

1

2(6 + 8− 4) = 5.

Let us denote such a form by ϕ0. By the definition, we have ϕ0 ⊂ ϕ. Sinceϕ0 ⊂ sc ⟨⟨u, v, k2⟩⟩, it follows that ϕ0 is a Pfister neighbor. Since ⟨⟨u, v, k2⟩⟩ =

8. MAIN THEOREM 79

ψ ⟨⟨k2⟩⟩, it follows that ⟨⟨u, v, k2⟩⟩F (ψ) is isotropic. Hence the Pfister neighbor

(ϕ0)F (ψ) of ⟨⟨u, v, k2⟩⟩F (ψ) is isotropic as well.

(2ii)⇒(2i). Let ϕ0 be a 5-dimensional Pfister neighbor such that ϕ0 ⊂ ϕ and(ϕ0)F (ψ) is isotropic. Let us write ϕ in the form ϕ = ϕ0 ⊥ ⟨s0⟩. Since ϕ0 is aPfister neighbor, there exists π ∈ GP3(F ) such that ϕ0 ⊂ π. We can write πin the form π = ϕ0 ⊥ −⟨s1, s2, s3⟩. Set γ = ⟨s0, s1, s2, s3⟩. We have

γ = ϕ− π ≡ ϕ = ⟨⟨a, b, c⟩⟩+ cρ ≡ cρ (mod I3(F )).

Since dim γ = dim cρ = 4 it follows from Wadsworth’s theorem ([86, Theorem7]) that γ is similar to cρ. Hence there exists k ∈ F ∗ such that γ = ckρ. Wehave

⟨⟨a, b, c⟩⟩ = ρ− cϕ = ρ− c(γ + π) = ρ− c(ckρ+ π) = ⟨⟨k⟩⟩ ρ− cπ.

Now it is sufficient to verify that ⟨⟨k⟩⟩ ρ ∈ I3(F (ρ)/F ) and π ∈ I3(F (ψ)/F ).We have ⟨⟨k⟩⟩ ρ = ⟨⟨a, b, c⟩⟩ + cπ ∈ I3(F ). Since dim(⟨⟨k⟩⟩ ρF (ρ))an < 8, theArason-Pfister Hauptsatz shows that ⟨⟨k⟩⟩ ρF (ρ) is hyperbolic. Thus ⟨⟨k⟩⟩ ρ ∈I3(F (ρ)/F ). Since ϕ0 ⊂ π and (ϕ0)F (ψ) is isotropic, πF (ψ) is isotropic as well.Since π ∈ GP3(F ), it follows that πF (ψ) is hyperbolic. Hence π ∈ I3(F (ψ)/F ).

Corollary 8.2. Let ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩ be an anisotropic quadraticform. Let ψ = ⟨−u,−v, uv, δ⟩ and ρ = ⟨−a,−b, ab, d⟩. Suppose that the groupCH2(Xψ×Xρ) is torsion-free. Then the following conditions are equivalent:

(1) ϕF (ψ) is isotropic;(2) there exits a 5-dimensional Pfister neighbor ϕ0 such that ϕ0 ⊂ ϕ and

(ϕ0)F (ψ) is isotropic

Proof. (1)⇒(2). By Item 1 of Proposition 8.1, we know that ⟨⟨a, b, c⟩⟩ ∈I3(F (ρ, ψ)/F ). Since TorsCH2(Xψ ×Xρ) = 0, Corollary 2.13 implies that

H3(F (ρ, ψ)/F ) = H3(F (ρ)/F ) +H3(F (ψ)/F )]; .

By Corollary 6.4, I3(F (ρ, ψ)/F ) ⊂ I3(F (ρ)/F )+ I3(F (ψ)/F )+I4(F ). Apply-ing Proposition 8.1 once again, we are done.(2)⇒(1). Obvious.

Lemma 8.3. Let ϕ be a 6-dimensional form and ψ be a 4-dimensionalform. Suppose that ψ is similar to a subform in ϕ. Then indC0(ϕ)⊗F C0(ψ) =1.

Proof. We can suppose that ψ ⊂ ϕ. Hence there exists a 2-dimensionalform µ such that ψ ⊥ µ = ϕ. Let E be a field extension of F generatedby√d± ϕ and

√d± ψ. Obviously ϕE, ψE ∈ I2(F ) and indC0(ϕ) ⊗F C0(ψ) =

indC0(ϕE) ⊗E C0(ψE). Thus we can reduce our problem to the case whereϕ, ψ ∈ I2(F ). Then µ ∈ I2(F ). Since dimµ = 2, the form µ is hyperbolic.Hence ϕ = ψ ⊥ H. Therefore C0(ϕ) = C0(ψ)⊗F M2(F ). Hence indC0(ϕ)⊗F

C0(ψ) = 1.


Corollary 8.4. Let ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩ be an anisotropic quadraticform. Let ψ = ⟨−u,−v, uv, δ⟩ and ρ = ⟨−a,−b, ab, d⟩.

Suppose that indC0(ϕ) ⊗F C0(ψ) = 1. Then the following conditions areequivalent:

(1) ϕF (ψ) is isotropic and the isotropy is standard;(2) there exits a 5-dimensional Pfister neighbor ϕ0 such that ϕ0 ⊂ ϕ and

(ϕ0)F (ψ) is isotropic;(3) ⟨⟨a, b, c⟩⟩ ∈ I3(F (ρ)/F ) + I3(F (ψ)/F ) + I4(F );(4) (a, b, c) ∈ H3(F (ρ)/F ) +H3(F (ψ)/F ).

Proof. (1)⇒(2). Let ϕ and ψ be such as in (1). Let us suppose that thecondition (2) is not satisfied. Then by the definition of standard isotropy, ψis similar to a subform of ϕ. By Lemma 8.3, we have indC0(ϕ)⊗F C0(ψ) = 1.This contradicts to our assumption.(2)⇒(1). Obvious.(3)⇐⇒(4)⇐⇒(1). Follows from Proposition 8.1 and Corollary 6.3.

Theorem 8.5. Let ϕ be an anisotropic 6-dimensional quadratic form andψ be a 4-dimensional quadratic form with d± ψ = d± ϕ = 1. Suppose thatϕF (ψ) is isotropic. Then there exits a 5-dimensional Pfister neighbor ϕ0 suchthat ϕ0 ⊂ ϕ and (ϕ0)F (ψ) is isotropic.

Proof. If indC0(ϕ) = 1 then ϕ is a Pfister neighbor. In this case wecan take ϕ0 to be equal to an arbitrary 5-dimensional subform in ϕ. In thecase indC0(ϕ) = 4, it follows from Theorem 5.5 of Chapter 3 that ϕF (ψ) isanisotropic and we have a contradiction. Thus we can assume that indC0(ϕ) =2. Then ϕ is similar to a form of the kind ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩. Since d± ψ =

d± ϕ, there exist u, v ∈ F ∗ such that ψ is similar to the form ⟨−u,−v, uv, d⟩.Replacing ϕ and ψ by similar forms, we can suppose that

ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩ and ψ = ⟨−u,−v, uv, d⟩ .Let ρ = ⟨−a,−b, ab, d⟩. It follows from Theorem 5.1 that TorsCH2(Xψ×Xρ) =0. Now the result required follows immediately from Corollary 8.2.

Proposition 8.6. Let ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩ and ψ = ⟨−u,−v, uv, δ⟩ beanisotropic quadratic forms. Suppose that indC0(ϕ) ⊗F C0(ψ) = 4. Then thefollowing conditions are equivalent:

(1) ϕF (ψ) is isotropic;(2) There is a 5-dimensional subform ϕ0 ⊂ ϕ which is a Pfister neighbor


Proof. Let ρ = ⟨−a,−b, ab, d⟩. Clearly C0(ϕ) =M2(F )⊗F C0(ρ). HenceindC0(ρ)⊗FC0(ψ) = 4. It follows from Theorem 5.8 that TorsCH2(Xρ×Xψ) =0. By Corollary 8.2, we are done.

Proposition 8.7. Let ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩ and ψ = ⟨−u,−v, uv, δ⟩ beanisotropic quadratic forms with δ /∈ F ∗2. Suppose that indC0(ϕ)⊗F C0(ψ) =1. Then the following conditions are equivalent:


(1) ϕF (ψ) is isotropic;(2) Either ψ is similar to a subform in ϕ or there exists a 5-dimensional

subform ϕ0 ⊂ ϕ which is a Pfister neighbor and (ϕ0)F (ψ) is isotropic.

Proof. (1)⇒(2). Since ϕ is anisotropic, we have d /∈ F ∗2. In view of The-orem 8.5 is is sufficient to consider the case dδ /∈ F ∗2. Let ρ = ⟨−a,−b, ab, d⟩.Since C0(ϕ) =M2(F )⊗F C0(ρ), we have indC0(ρ)⊗F C0(ψ) = 1. Thus all theassumptions of §7 hold. Propositions 7.10 and 8.1 show that at least one ofthe following conditions holds:

1) ⟨⟨a, b, c⟩⟩ ∈ I3(F (ρ)/F ) + I3(F (ψ)/F ) + I4(F ),2) ⟨⟨a, b, c⟩⟩ ∈ Γ(ρ, ψ) + I4(F ).

In the first case, Proposition 8.1 asserts that there exists a 5-dimensional sub-form ϕ0 ⊂ ϕ which is a Pfister neighbor and (ϕ0)F (ψ) is isotropic.

Thus we can suppose that ⟨⟨a, b, c⟩⟩ ∈ Γ(ρ, ψ) + I4(F ). Let γ = l1ρ+ l2ψ +⟨⟨dδ⟩⟩ ∈ Γ(ρ, ψ) be such that ⟨⟨a, b, c⟩⟩ ∈ γ + I4(F ). Since ⟨⟨a, b, c⟩⟩ = ρ − cϕ,we have

l1ρ− l1cϕ = l1 ⟨⟨a, b, c⟩⟩ ≡ ⟨⟨a, b, c⟩⟩ ≡ γ = l1ρ+ l2ψ + ⟨⟨dδ⟩⟩ (mod I4(F )).

Hence l1cϕ + l2ψ + ⟨⟨dδ⟩⟩ ∈ I4(F ). Since dim(l1cϕ + l2ψ + ⟨⟨dδ⟩⟩)an ≤ 6 + 4 +2 = 12 < 16, the Arason-Pfister Hauptsatz shows that l1cϕ + l2ψ + ⟨⟨dδ⟩⟩ =0. Therefore ϕ = −cl1l2ψ − cl1 ⟨⟨dδ⟩⟩. Since dimϕ = 6 = dim(−cl1l2ψ ⊥−cl1 ⟨⟨dδ⟩⟩), we have ϕ = −cl1l2ψ ⊥ −cl1 ⟨⟨dδ⟩⟩. Hence ψ is similar to a subformin ϕ.(2)⇒(1). Obvious.

Together with results described in Introduction, Theorem 8.5, Propositions8.6 and 8.7 give rise to the following

Theorem 8.8. Let ϕ be an anisotropic quadratic form of dimension ≤ 6and ψ be such that ϕF (ψ) is isotropic. If the isotropy is non-standard then

• dimϕ = 6 and dimψ = 4;• 1 = d± ϕ = d± ψ = 1;• indC0(ϕ) = 2; and• indC0(ϕ)⊗F C0(ψ) = 2.

9. The case of index 2

Theorem 8.8 implies that if there exists a quadratic form ϕ of dimension≤ 6 having a non-standard isotropy over the function field of a quadraticform ψ, then there are a, b, c, d, u, v, δ ∈ F ∗ such that ϕ ∼ ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩,ψ ∼ ⟨−u,−v, uv, δ⟩, d, δ, dδ /∈ F ∗2, and ind((a, b)⊗F (u, v))F (

√d,√δ) = 2.

Set ρ = ⟨−a,−b, ab, d⟩. By Corollary 8.2, if TorsCH2(Xψ ×Xρ) = 0, thenthe isotropy is standard.

In this section we prove the following


Theorem 9.1. Let a, b, u, v, d, δ ∈ F ∗2 be such that d, δ, dδ /∈ F ∗2. Let ρ =⟨−a,−b, ab, d⟩ and ψ = ⟨−u,−v, uv, δ⟩. Suppose that indC0(ρ)⊗F C0(ψ) = 2.The following conditions are equivalent:

(1) TorsCH2(Xρ ×Xψ) = 0;(2) there exists c ∈ F ∗ such that the quadratic form ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩ is

isotropic over F (ψ), but the isotropy is not standard.

Proof. (2)⇒(1). Obvious in view of Corollary 8.2.(1)⇒(2). Since TorsCH2(Xρ × Xψ) = 0, it follows from Corollary 2.13 thatthere exists w ∈ H3(F (ρ, ψ)/F ) such that w /∈ H3(F (ρ)/F ) + H3(F (ψ)/F ).Let ρ0 = ⟨−a,−b, ab⟩. It follows from Theorem 5.9 that indC0(ρ0)⊗F C0(ψ) =indC0(ρ) ⊗F C0(ψ) = 2. Therefore indC0(ρ0) ⊗F C0(ψ) = 4. By The-orem 5.8, we have TorsCH2(Xρ0 × Xψ) = 0. By Corollary 2.13, we haveH3(F (ρ0, ψ)/F ) = H3(F (ρ0)/F ) +H3(F (ψ)/F ). Hence

w ∈ H3(F (ρ, ψ)/F ) ⊂ H3(F (ρ0, ψ)/F ) = H3(F (ρ0)/F ) +H3(F (ψ)/F ).

Since H3(F (ρ0)/F ) = (a, b) ∪ H1(F ), there exists c ∈ F ∗ such that w −(a, b, c) ∈ H3(F (ψ)/F ), i.e. w ≡ (a, b, c) (mod H3(F (ψ)/F )). By the assump-tion on w, we see that (a, b, c) ∈ H3(F (ρ, ψ)/F ) and (a, b, c) /∈ H3(F (ρ)/F ) +H3(F (ψ)/F ). Therefore, ⟨⟨a, b, c⟩⟩ ∈ I3(F (ρ, ψ)/F and

⟨⟨a, b, c⟩⟩ /∈ I3(F (ρ)/F ) + I3(F (ψ)/F ) +H4(F ) .

By Proposition 8.1, the quadratic form ϕF (ψ) is isotropic. By Corollary 8.4,the isotropy is not standard.

CHAPTER 5

Some new examples in the theory of quadratic forms

We construct a 6-dimensional quadratic form ϕ and a 4-dimensional quadraticform ψ over some field F such that ϕ becomes isotropic over the function fieldF (ψ) but every proper subform of ϕ is still anisotropic over F (ψ). It is anexample of non-standard isotropy with respect to some standard conditionsof isotropy for 6-dimensional forms over function fields of quadrics, knownpreviously.

Besides of that, we produce an example of 8-dimensional quadratic form ϕwith trivial determinant such that the index of the Clifford invariant of ϕ is 4but ϕ can not be represented as a sum of two 4-dimensional forms with trivialdeterminants. Using this, we find a 14-dimensional quadratic form with trivialdiscriminant and Clifford invariant which which is not similar to a differenceof two 3-fold Pfister forms.


0. Introduction

Let F be a field of characteristic = 2. An important problem in the al-gebraic theory of quadratic forms is to classify pairs of anisotropic quadraticforms ϕ, ψ such that ϕF (ψ) is isotropic, where F (ψ) is the function field of ψ,i.e. the function field of the projective quadric determined by ψ. In the casedimϕ ≤ 5, a complete classification is known (see [15]). The case dimϕ = 6was studied in [16], [44], [45], [49], and [54]. In the case where dimϕ = 6 anddimψ = 4, a complete classification was obtained. It was shown in Chapters3 and 4, that the same classification is valid for 4-dimensional forms ψ, if thecase where dimϕ = 6, dimψ = 4, 1 = det± ϕ = det± ψ = 1, indC0(ϕ) = 2, andindC0(ϕ)⊗ C0(ψ) = 2 is excluded. Here (see Section 15) we construct in thisexcepted case an example of ϕ and ψ with non-standard (i.e. not matchingthe old classification) isotropy of ϕ over F (ψ) (see Theorem 15.2). It is possi-ble to explain what this “non-standard isotropy” does exactly mean withoutdescribing the old classification (Lemma 15.3): isotropy of ϕ over F (ψ) is non-standard if and only if the form ϕ is F (ψ)-minimal, i.e. no proper subformof ϕ becomes isotropic over F (ψ). A stronger version of Theorem 15.2 statesthat an example of non-standard isotropy can be obtained starting from anarbitrary anisotropic 4-dimensional form ψ (with detψ = 1) over an arbitraryfield F0 by passing to an appropriate extension F of F0 (see Corollary 15.4).

Let I(F ) be the ideal of even-dimensional forms in the Witt ring W (F ).Another important problem in the algebraic theory of quadratic forms is to

83

84 5. NEW EXAMPLES OF QUADRATIC FORMS

give a classification of low-dimensional quadratic forms belonging to In(F ) fora fixed n > 0. For n = 2 and for n = 3, this problem was studied by manyauthors. In [28] Jacobson proved that quadratic forms ϕ ∈ I2(F ) of dimension≤ 6 are uniquely determined up to similarity by the Clifford invariant c(ϕ).There exists a good description of 8-dimensional forms ϕ ∈ I2(F ) satisfyingthe condition indC(ϕ) ≤ 2. Namely, these quadratic forms can be written astensor product of a 2-dimensional subform and a 4-dimensional subform (seee.g. [41, Example 9.12]). The case of 8-dimensional quadratic forms ϕ ∈ I2(F )with indC(ϕ) = 4 is much more complicated. It was an open question if thesequadratic forms can be written as τ1 ⊥ τ2, where τ1 and τ2 are 4-dimensionalforms with trivial determinant. In Section 13 we construct a counterexamplefor this question (Corollary 13.8). Nevertheless we find a “weak version” of thedecomposition ϕ = τ1 ⊥ τ2. Note that quadratic forms of the type τ1 ⊥ τ2 canbe regarded as Scharlau’s transfer sL/F (τ) in the degenerate case L = F × F .We show that an arbitrary 8-dimensional form ϕ ∈ I2(F ) with indC(ϕ) = 4can be represented as Scharlau’s transfer sL/F (τ), where L/F is an (etale)quadratic extension and τ is a 4-dimensional L-form with trivial determinant(see Theorem 13.10).

In Section 14 we study quadratic forms ϕ ∈ I3(F ). The structure of ϕin the case dimϕ ≤ 12 was described by Pfister in [68, Satz 14 und Zusatz](see also [18]). Our aim is to study 14-dimensional quadratic forms in I3(F ).In [72] M. Rost proved that an arbitrary 14-dimensional quadratic form can

be represented (up to similarity) as Scharlau’s transfer sL/F (√d τ ′), where

L = F (√d) and τ ′ is the pure subform of a 3-fold Pfister form. Note that

in the degenerate case L = F × F we get the decomposition ϕ = k(τ ′1 ⊥−τ ′2), where τ ′1, τ ′2 are pure subforms of 3-fold Pfister forms τ1, τ2 and k ∈F ∗. It was an open question if any 14-dimensional form ϕ ∈ I3(F ) can bewritten in the form ϕ = k(τ ′1 ⊥ −τ ′2). It was remarked by D. Hoffmann (1995,Bielefeld, oral communication) that this question is equivalent to the discussedabove question on 8-dimensional forms ϕ ∈ I2(F ) with indC(ϕ) = 4. Usingthe counterexample for 8-dimensional forms, we construct (in Section 14) acounterexample for 14-dimensional forms.

Similar counterexamples of 8-dimensional and 14-dimensional forms in thecase of characteristic 0 are independently constructed in [21] by using a com-pletely different technique.

Now we explain the structure of the Chapter. It can be divided into twoparts: Sections 2–11 and Sections 12–15. All main results listed above areobtained in the second part. In the first part, the necessary preparations aremade. The results of Sections 8 and 10 are needed for the counterexamplesof 8-and-14-dimensional forms, while the results of Sections 9 and 11 are forthe non-standard isotropy. The constructions and proofs of Sections 8, 10 areparallel to that of Sections 9, 11. It is the reason for why we included the re-sults on 8,14-dimensional forms and on the non-standard isotropy in the same


Chapter. However, Sections 8, 10 and Sections 9, 11 are written in an indepen-dent manner, so that the reader interested only in one of two groups of resultscan choose (although, in order to understand more complicated calculations ofSection 9, it is better to go through Section 8 first).

Now we explain more precisely the contents of several first preparationSections.

In Section 2, we show that certain products of (generalized) Severi-Brauervarieties considered as schemes over certain subproducts via the projection canbe naturally identified with grassmanians (Corollary 2.4). It was already donein Corollary 6.4 of Chapter 1 and in Proposition 5.3 of Chapter 2. Howeverthis time we need more explicit information: namely, we need a description ofthe vector bundle on the product of the Severi-Brauer varieties correspondingto the tautological vector bundle on the grassmanian under that identification.The answer is given in terms of the canonical vector bundles on the Severi-Brauer varieties.

Let Γ be the grassmanian of “n-planes” in a vector bundle on a varietyX. In Section 3, we describe the Grothendieck group of Γ together with thetopological filtration on it in terms of the Grothendieck group of X. Thegeneral answer (Proposition 3.3) is an easy consequence of the well-knownresult on the Chow group of a grassmanian. Some additional negligible effortsare made to formulate the result in terms of characteristic classes of the class[T ] of the tautological vector bundle T ; classically ([12, Proposition 14.6.5])characteristic classes of −[T ] are used, what is not convenient for practical use.After all this, we apply the general assertion to the case of the grassmanian of2-planes in a rank 4 vector bundle and get some explicit formulas which arethen used in Sections 8 and 9.

In Section 4, we reprove that the pull-back to the generic fiber of a flatmorphism is surjective. For what kind of groups? Well, our final goal is thetopological filtration i.e., each term of that (Corollary 4.3). We reach the goalstarting with the Chow groups (Proposition 4.1) and passing after that to thequotients of the topological filtration (Corollary 4.2). The statement on theChow groups is not new and even was already used several times in this work.It is a formal consequence of the spectral sequence [39, Theorem 3.1]. Here,we give a short direct proof or, better to say, an explanation of the evidenceof this fact (Proposition 4.1).


1.1. Quadratic forms. By ϕ ⊥ ψ, ϕ ≃ ψ, and [ϕ] we denote respectivelyorthogonal sum of forms, isometry of forms, and the class of ϕ in the Witt ringW (F ) of the field F . To simplify notation we write ϕ1+ϕ2 instead of [ϕ1]+[ϕ2].The maximal ideal of W (F ) generated by the classes of the even-dimensionalforms is denoted by I(F ). The anisotropic part of ϕ is denoted by ϕan. Wedenote by ⟨⟨a1, . . . , an⟩⟩ the n-fold Pfister form ⟨1,−a1⟩ ⊗ · · · ⊗ ⟨1,−an⟩ andby Pn(F ) the set of all n-fold Pfister forms. The set of all forms similar to an


n-fold Pfister form we denote by GPn(F ). For any field extension L/F , we putϕL = ϕ⊗F L.

For a quadratic extension L/F and an L-form ϕ, we denote by sL/F (ϕ)Scharlau’s transfer [76, Ch. 2, §5] corresponding to the F -linear homomor-

phism 12TrL/F : L → F . In the case where L = F (

√d), we have sL/F (⟨1⟩) =

⟨1, d⟩ and sL/F (⟨√d ⟩) = ⟨1,−1⟩.

For a quadratic form ϕ of dimension ≥ 3, we denote by Xϕ the projectivevariety given by the equation ϕ = 0. We set F (ϕ) = F (Xϕ).

1.2. Linked forms. We say that quadratic F -forms ϕ and ψ are linked ifthe following equivalent conditions hold:

• there exists a 2-dimensional form µ which is similar to a subform of ϕand to a subform of ψ,

• there exists a field extension L/F of degree ≤ 2 such that ϕL and ψLare isotropic,

If ϕ and ψ are forms of dimension ≥ 3, then the condition that ϕ and ψare linked can be reformulated as follows: there exists a closed point of degree≤ 2 on the variety Xϕ ×Xψ.

1.3. K-theory and Chow groups. For a smooth algebraic F -varietyX, its Grothendieck ring is denoted by K(X). This ring is equipped with thefiltration by codimension of support (which respects the multiplication). Fora ring (or a group) with filtration A, we denote by G∗A the adjoint gradedring (resp., the adjoint graded group). There is a canonical surjective homo-morphism of the graded Chow ring CH∗(X) onto G∗K(X), its kernel consistsonly of torsion elements and is trivial in the 0-th, 1-st, and 2-nd graded com-ponents ([81, §9]). For a geometrically integral variety of dimension d we setCHi(X) = CHd−i(X) and GiK(X) = Gd−iK(X).

1.4. Algebras. Let A be an algebra over a field F . For a field extensionE/F (or, more generally, for a unital commutative F -algebra E), we denoteby AE the E-algebra A⊗F E. For an F -variety X (or, more generally, for anF -scheme X), we denote by AX the constant X-sheaf of algebras given by A.

In Section 2, the category of commutative unital F -algebras is denoted byF -alg.

2. Products of Severi-Brauer varieties

Let F be a field and let A be a central simple algebra over F .

Let n ≥ 0. The generalized Severi-Brauer variety Ydef= SB(n,A) of A

is characterized as follows: for any R ∈ F -alg, the set of R-points Y (R)def=

MorF (SpecR, Y ) of the variety Y consists of the right ideals J of the Azumaya

R-algebra ARdef= A⊗F R having two following properties:

• the injection of AR-modules J ↪→ AR splits (in particular, J is projectiveas the R-module);

2. PRODUCTS OF SEVERI-BRAUER VARIETIES 87

• rk J = n, where rk J is the R-rank of J divided by degA;

moreover, for any homomorphism R → R′ in the category F -alg, the mapY (R) → Y (R′) is given by tensor multiplication J 7→ J ⊗R R

′.The (usual) Severi-Brauer variety SB(A) of A is by definition the variety

SB(1, A).

Example 2.1. Let A be a quaternion algebra (a, b), where a, b ∈ F ∗. TheSeveri-Brauer variety SB(A) is isomorphic to the projective conic determinedby the quadratic form ⟨1,−a,−b⟩.

Example 2.2. Let A be a biquaternion algebra (a1, b1)⊗(a2, b2), where a1,b1, a2, b2 ∈ F ∗. The generalized Severi-Brauer variety SB(2, A) is isomorphicto the projective quadric determined by the Albert form

⟨−a1,−b1, a1b1, a2, b2,−a2b2⟩ .

The canonical vector bundle J on the generalized Severi-Brauer variety Yis defined as follows: for any R ∈ F -alg and a point J ∈ Y (R), the fiber ofJ over J is the R-module J ; if R → R′ is a homomorphism in F -alg, then

the map of the fibers J → J ′, where J ′ def= J ⊗R R

′ ∈ Y (R′), is defined by theformula x 7→ x⊗ 1.

Since every fiber of J is right ideal, J has a structure of right AY -module.

Proposition 2.3. Let Xdef= SB(A), Y

def= SB(n,Aop); let I and J be the

canonical vector bundles on X and Y . The product X ×Y , considered over Xvia the first projection, can be naturally identified (as a scheme over X) withthe grassmanian IΓn(I) of n-planes in I; by this identification, the tautologicalvector bundle on the grassmanian corresponds to a vector bundle on X × Yisomorphic to I ⊗AX×Y J .

Proof. Let R ∈ F -alg and let I be an R-point of X. To prove the firstpart of Proposition, it suffices to describe a natural bijection of the fibersover I. The fiber of X × Y over the point I is the set Y (R). The fiber ofIΓn(I) over the point I is the set of R-submodules N of the R-module I suchthat the injection N ↪→ I splits and rkRN = n. For any N like that, the setHomR(I,N) is a right ideal of the R-algebra EndR I = Aop

R and thus determinesan element of Y (R). In this way, we get the natural bijection required.

To describe an isomorphism of the vector bundles (mentioned in the secondstatement of Proposition), it suffices to give a natural isomorphism of the R-modules I⊗ARHomR(I,N) and N . This is given by the rule x⊗f 7→ f(x).

Now, let A1, . . . , Am, A be central simple F -algebras such that

A = A⊗i11 ⊗ · · · ⊗ A⊗im

m with certain i1, . . . , im ≥ 0 .

Denote by X1, . . . , Xm the Severi-Brauer varieties of the algebras A1, . . . , Am.

Put Xdef= X1 × · · · × Xm and Y

def= SB(n,Aop). For every j = 1, . . . ,m, let

Ij be the canonical vector bundle on Xj. Put I def= I⊗i1

1 ⊗ · · · ⊗ I⊗imm ; it is


a right AX-module. Let J be the canonical vector bundle on Y ; it is a leftAY -module.

Corollary 2.4. In the notation introduced right above, the product X ×Y , considered over X via the first projection, can be naturally identified (as ascheme over X) with the grassmanian IΓn(I); by this identification, the tau-tological vector bundle on the grassmanian corresponds to a vector bundle onX × Y isomorphic to I ⊗AX×Y J .

Proof. Put X ′ def= SB(A) and let I ′ be the canonical vector bundle on X ′.

Let R ∈ F -alg. An element of the set X(R) = X1(R) × · · · ×Xm(R) is acollection (I1, . . . , Im) where Ij is a right ideal in (Aj)R. The tensor productI1 ⊗R · · · ⊗R Im is a right ideal of AR. In this way we get a map of setsX(R) → X ′(R) which is natural with respect to R. Let X → X ′ be thecorresponding morphism of F -varieties.

Consider the cartesian square

X × Y −−−→ X ′ × Yy yX −−−→ X ′ .

By Proposition, we can identify the product X ′ × Y with the grassmanianIΓn(I ′). Since the inverse image of the vector bundle I ′ with respect to themorphism X → X ′ is I, the variety X × Y can be therefore identified withIΓn(I).

Taking the inverse of the vector bundle I ′ ⊗AX′×YJ with respect to the

morphism X × Y → X ′ × Y , we get the vector bundle I ⊗AX×Y J . It provesthe second statement of Corollary.

3. The Grothendieck group of a grassmanian

By ring we mean a commutative unital ring.Let R be a ring. We consider only the descending filtrations R(i) (i ∈ Z)

on R satisfying the following conditions:

• R(i) ·R(j) ⊂ R(i+j) for all i, j and• R(0) = R.

Let R be a ring with filtration and let M be an R-module. We consideronly the descending filtrations M (i) on M satisfying the following conditions:

• R(i) ·M (j) ⊂M (i+j) for all i, j and• M (0) =M .

Definition 3.1. Let R be a ring with filtration, letM be an R-module, lete1, . . . , ek ∈M , and let α1, . . . , αk ∈ Z. We define the filtration on M inducedby the conditions ei ∈M (αi) for i = 1, . . . , k to be the smallest filtration on Msatisfying these conditions.

The following assertion is evident:

3. THE GROTHENDIECK GROUP OF A GRASSMANIAN 89

Lemma 3.2. For every n ≥ 1, the n-th term M (n) of the filtration on Minduced by the conditions ei ∈M (αi) (i = 1, . . . , k) is determined by the formula

M (n) =k∑j=1

R(n−αj) · ej .

We fix the following notation for the rest of this Section: F is an arbitraryfield, X is a smooth F -variety, r ≥ n ≥ 0 are integers, E is a rank r vector

bundle over X, Γdef= IΓn(E) is the grassmanian of n-planes in the vector bundle

E , and π is the structure morphism Γ → X. We put m = r − n.We denote by T the tautological vector bundle on Γ (also called the universal

vector subbundle on Γ, see [12, §14.6]). The rank of T equals n.An (m,n)-partition λ is a sequence of integers (λ1, . . . , λm) of length m

satisfying the condition n ≥ λ1 ≥ · · · ≥ λm ≥ 0. The weight |λ| of λ is bydefinition the sum λ1 + · · ·+ λm.

Let sdef= (s1, s2, . . . ) be a sequence of variables. Additionally, we put s0

def= 1

and sidef= 0 for all i < 0. For an (m,n)-partition λ, the Schur polynomial ∆λ(s)

of λ is the determinant of the matrix (sλi+j−i)mi,j=1 (see also the definition of

[12, §14.5]). It is a homogeneous polynomial of weight |λ|, if every si is takenwith the weight i.

For every i ≥ 1, let us substitute si = (−1)ici([T ]) where [T ] is the class ofT in K(Γ) and ci : K(Γ) → K(Γ) is the i-th Chern class with values in K (see

Definition 2.1 of Chapter 1). For any (m,n)-partition λ, we put ∆λdef= ∆λ(s).

Since ci([T ]) ∈ K(Γ)(i), we have ∆λ ∈ K(Γ)(|λ|).We consider K(Γ) as a K(X)-module via the pull-back homomorphism

π∗ : K(X) → K(Γ).

Proposition 3.3. The K(X)-moduleK(Γ) is free and the elements {∆λ}λ,where λ runs over all (m,n)-partitions, form its basis. The topological filtrationon K(Γ) is induced by the conditions ∆λ ∈ K(Γ)(|λ|) (see Definition 3.1).

Proof. We have to show that the map⊕λK(X)

∑·∆λ−−−→ K(Γ)

is an isomorphism of groups with filtrations, where the direct sum is taken overall (m,n)-partitions λ and for every λ, the λ-summand K(X) of the direct sumis considered with the topological filtration shifted by |λ|.

Since the filtrations are finite, it suffices to show that the induced homo-morphism of adjoint graded groups⊕

λ

G∗−|λ|K(X)∑

·∆λ−−−→ G∗K(Γ)

is an isomorphism, where ∆λ the class of ∆λ in the quotient G|λ|K(Γ). Weprove it (see Proposition 3.7) after we have introduced some additional notationand proved some preliminary assertions.


We repeat that we denote by ∆λ the class of ∆λ in the quotient G|λ|K(Γ).Note that ∆λ can be also defined directly in the similar way as ∆λ by usingthe the Chern classes with values in G∗K (see Definition 2.8 of Chapter 1)instead of the Chern classes with values in K.

Let us now substitute si = ci(−[T ]) ∈ K(Γ) for every i ≥ 1. For an (n,m)-

partition λ′, we put ∇λ′def= ∆λ′(s). We have ∇λ′ ∈ K(Γ)(|λ

′|) and we denote by∇λ′ ∈ G|λ′|K(Γ) the residue class. Note that ∇λ′ can be also defined directlyin the similar way as ∇λ′ by using the the Chern classes with values in G∗K.Now, taking the Chern classes with values in CH∗, let us define in the similarway one more element ∇CH

λ′ ∈ CH|λ′|(Γ).

Proposition 3.4. The map⊕λ′

CH∗−|λ′|(X)

∑·∇CHλ′−−−−→ CH∗(Γ),

where λ′ runs over the set of all (n,m)-partitions, is an isomorphism of gradedgroups.

Proof. First of all, it is evidently a homomorphism of graded groupsbecause ∇CH

λ′ ∈ CH|λ′|(Γ) for every λ′. Thus we only have to show that it is anisomorphism of groups, without taking care for the gradations. This is donein [12, Proposition 14.6.5].

An (n,m)-partition λ′ is called dual to an (m,n)-partition λ, if λ′i (for everyi = 1, . . . , n) is equal to the quantity of λ1, . . . , λm which are ≥ i.

Lemma 3.5. Let λ be an (m,n)-partition and let λ′ be the (n,m)-partitiondual to λ. Then ∆λ = ∇λ′. In particular, ∆λ = ∇λ′.

Proof. The first relation follows from [12, Lemma 14.5.1]. After that thesecond relation is evident (note that |λ| = |λ′|).

Lemma 3.6 (“Duality theorem”). Let λ′ and λ′′ be (n,m)-partitions suchthat |λ′′|+ |λ′| ≤ nm and let α ∈ G∗K(X). Then

π∗(∇λ′ · ∇λ′′ · π∗(α)

)=

{α, if λ′′i = m− λ′n−i+1 for all 1 ≤ i ≤ n;

0, otherwise.

Proof. Denote by α ∈ CH∗(X) an arbitrary preimage of α with respect tothe canonical epimorphism CH∗(X) � G∗K(X). By the Chow group variantof the duality theorem ([12, Proposition 14.6.3]), we have the formula

π∗(∇CHλ′ · ∇CH

λ′′ · π∗(α))=

{α, if λ′′i = m− λ′n−i+1 for all 1 ≤ i ≤ n;

0, otherwise.

Since the canonical epimorphism CH∗(−) � G∗K(−) commutes with push-forwards, pull-backs, and Chern classes, the formula required follows.

3. THE GROTHENDIECK GROUP OF A GRASSMANIAN 91

Proposition 3.7. The map⊕λ

G∗−|λ|K(X)∑

·∆λ−−−→ G∗K(Γ)

is an isomorphism of graded groups.

Proof. By Lemma 3.5, the homomorphism∑

·∆λ coincides with the ho-momorphism

∑·∇λ′ , where λ

′ runs over all (n,m)-partitions. By Proposition3.4, the upper arrow in the commutative diagram⊕

λ′

CH∗(X)

∑·∇CHλ′−−−−→ CH∗(Γ)y y⊕

λ′

G∗K(X)∑

·∇λ′−−−−→ G∗K(Γ)

is an isomorphism. Therefore the bottom arrow, i.e. the homomorphism∑·∇λ′ , is surjective. So, it remains to prove injectivity.We prove injectivity of

∑·∇λ′ in exactly the same way as injectivity of∑

·∇CHλ′ is proved in [12] (see the beginning of the proof of [12, Proposition

14.6.5]). Suppose that the homomorphism is not injective. Let ⊕αλ′ be a

non-zero element of ker(∑

·∇λ′). Choose an (n,m)-partition λ′ of maximal

weight such that αλ′ = 0. Define another (n,m)-partition λ′′ as follows: λ′′idef=

m− λ′n−i+1 for i = 1, . . . , n. By Lemma 3.6, we have

0 = π∗(∇λ′′ · (

∑λ′

·∇λ′)(α))=∑λ′π∗(∇λ′′ · ∇λ′ · π

∗(α))= αλ′ ,

a contradiction.

The proof of Proposition 3.3 is complete.

Remark 3.8. The assertion that K(Γ) is a free K(X)-module holds in amore general situation, namely in the case where Γ is a twisted grassmanianover X. A proof can be found in [50, Theorem 4.4]. However the systemof generators which appears there has no “good relation” to the topologicalfiltration and differs from that of Proposition 3.3.

We are especially interested in the case where m = n = 1 (i.e. in the caseof a projective line bundle) and in the case where m = n = 2.

Corollary 3.9. Let Γ → X be a projective line bundle and let T be thetautological vector bundle on Γ. Then K(Γ) is a free K(X)-module with thebasis 1, 1 − [T ]; the topological filtration on K(Γ) is induced by the condition1− [T ] ∈ K(Γ)(1).

Proof. We apply Proposition 3.3 to the particular situation of Corollary.Since nowm = n = 1, we have only two (m,n)-partitions: λ = (0) and λ = (1).In the first case we get ∆λ = 1, in the second case we get ∆λ = 1− [T ].


Corollary 3.10. Let Γ → X be the Grassmanian of 2-planes in a rank4 vector bundle over X. Denote by −η, µ ∈ K(Γ) respectively the first andthe second Chern classes of the tautological vector bundle on Γ. Then K(Γ) isa free K(X)-module with the basis 1, η, µ, η2, µη, µ2; moreover, the topologicalfiltration on K(Γ) is induced by the conditions η ∈ K(Γ)(1); η2, µ ∈ K(Γ)(2);ηµ ∈ K(Γ)(3); µ2 ∈ K(Γ)(4).

Proof. We apply Proposition 3.3 to the particular situation of Corollary.We are going to compute ∆λ for every (2, 2)-partition λ. We use the notationintroduced above. In particular, s0, s1, s2 are variables and s−1 = 0 = s3.

• For λ = (2, 2), we have

∆λ(s) = det

(s2 s3s1 s2

)= s22 − s1s3 = s22 ; therefore ∆(2,2) = µ2.


∆λ(s) = det

(s2 s3s0 s1

)= s2s1 − s0s3 = s2s1 ; therefore ∆(2,1) = µη.


∆λ(s) = det

(s2 s3s−1 s0

)= s2s0 − s−1s3 = s2s0 ; therefore ∆(2,0) = µ.


∆λ(s) = det

(s1 s2s0 s1

)= s21 − s0s2 ; therefore ∆(1,1) = η2 − µ.


∆λ(s) = det

(s1 s2s−1 s0

)= s1s0 − s−1s2 = s1s0 ; therefore ∆(1,0) = η.


∆λ(s) = det

(s0 s1s−1 s0

)= s20 − s−1s1 = s20 ; therefore ∆(0,0) = 1.

To finish the proof, we just replace ∆(1,1) by ∆(1,1) +∆(2,0) = η2.

4. Pull-back to generic fiber

We fix the following notation for this Section: F is a field, X and Y areirreducible varieties over F , π : X → Y is a flat morphism, θ is the generic

point of Y , and Xθdef= X×Y SpecF (θ) is the fiber of π over θ. We are going to

consider the pull-back with respect to the flat morphism of schemes i : Xθ → X.Note that from the set-theoretical (even topological) point of view, Xθ is

really the fiber of π over the point θ (see [14, Exercise 3.10 after §3 of ChapterII]). In particular, Xθ is a subset of X.

5. WEIL TRANSFER VIA GALOIS DESCENT 93

The group CH∗(X) is generated by the classes [x] of points x ∈ X. Thepull-back homomorphism i∗ : CH∗(X) → CH∗(Xθ) is determined by the fol-lowing rule: if x ∈ Xθ (i.e., if π(x) = θ), then i∗([x]) = 0; if x ∈ Xθ (i.e., ifπ(x) = θ), then i∗([x]) = [x] ∈ CH∗(Xθ).

Proposition 4.1. The pull-back homomorphism i∗ : CH∗(X) → CH∗(Xθ)is surjective.

Proof. Take any generator αdef= [x] of the group CH∗(Xθ), where x ∈ Xθ.

If we consider x as a point of X, we get an element βdef= [x] ∈ CH∗(X) such

that i∗(β) = α.

Corollary 4.2. The pull-back homomorphism i∗ : G∗K(X) → G∗K(Xθ)is surjective.

Proof. The diagram

CH∗(X) −−−→ CH∗(Xθ)y yG∗K(X) −−−→ G∗K(Xθ)

where the vertical arrows are the canonical epimorphisms (see Section 1), iscommutative. Since the map CH∗(X) → CH∗(Xθ) is surjective (Proposition4.1) and the map CH∗(Xθ) → G∗K(Xθ) is surjective, the map G∗K(X) →G∗K(Xθ) is surjective as well.

Corollary 4.3. For any n ≥ 0, the pull-back homomorphism

i∗ : K(X)(n) → K(Xθ)(n)

is surjective.

Proof. Follows from Corollary 4.2.

5. Weil transfer via Galois descent

In this Section, L/F is a finite Galois field extension of degree n with theGalois group G. All varieties are assumed to be quasi-projective.

Definition 5.1. Let X be an F -variety. An L/F -form of X is an F -variety Y supplied with an isomorphism YL→XL. A morphism of an L/F -formY to another L/F -form Y ′ of the same variety X is a morphism of F -varietiesf : Y → Y ′ such that the diagram of L-morphisms

YLfL−−−→ Y ′

L↘ ↙

XL

commutes.


Let X be an F -variety. The (abstract) group Aut(XL) of the automor-phisms of the L-variety XL can be supplied with a structure of G-module inthe standard way (see [79, §1.1 de Chapitre III]): for τ ∈ G and f ∈ Aut(XL)

one puts τ(f)def= (idX ⊗ τ) ◦ f ◦ (idX ⊗ τ−1) where idX ⊗ τ is the automor-

phism of the scheme XL over F given by τ . Denote by Z1(G,Aut(XL)) =Z1(L/F,Aut(XL)) the set of 1-cocycles on G with values in Aut(XL) ([79,§5.1 de Chapitre I]).

Any L/F -form Y of X determines a cocycle z ∈ Z1(L/F,Aut(XL)) ([79,§1.3 de Chapitre III]): for any τ ∈ G, the automorphism zτ ∈ Aut(XL) is thecomposition

XL→YLidY ⊗τ−−−→ YL→XL

idX⊗τ−1

−−−−−→ XL .

Moreover, the rule described above is a 1-1-correspondence between the set ofL/F -forms of X (up to the canonical isomorphism) and the set

Z1(L/F,Aut(XL))

(compare to [79, 1.3 de Chapitre III]).Now suppose that X =

∏G T (the product of n copies of T numbered by

the elements of G), where T is a variety over F . We are going to construct aspecial 1-cocycle z ∈ Z1(L/F,Aut(XL)) in this special situation.

The group Sn of the permutations of the set G (recall that n = |G|) can benaturally identified with a subgroup of the group AutXL: a permutation ofthe set G corresponds to the automorphism of the product XL =

∏G YL given

by the permutation of the factors. Moreover, this way Sn is a G-submoduleof AutXL with trivial action of G. In particular, the set Z1(G,Sn) consists ofthe group homomorphisms G→ Sn.

For any τ ∈ G denote by zτ ∈ Sn the left translation by τ , that is thepermutation σ 7→ τσ of the set G. The map z : G → Sn given by the ruleτ 7→ zτ is a group homomorphism and thus z ∈ Z1(G,Sn).

We shall consider z as an element of Z1(L/F,Aut(XL)).

Definition 5.2. The following data are fixed: a finite Galois field exten-sion L/F and an F -variety T . The L/F -form (see Definition 5.1) of the variety

XLdef=∏

G TL determined by the cocycle z ∈ Z1(L/F,Aut(XL)) constructedabove will be denoted by R(T ) or RL/F (T ).

Remark 5.3. The variety RL/F (T ) is the same as the Weil transfer (see[8, 6.6 de §1 de Chapitre I] and/or [77, Chapter 4]) of the L-variety TL withrespect to the extension L/F . Usually, working with varieties over fields, onedefines the Weil transfer for any finite separable field extension L/F and aquasi-projective L-variety. However, we are interested here only in the casewhere the extension L/F is Galois and the L-variety “comes from F”. Defi-nition 5.2 can be regarded as an alternative definition of the Weil transfer inthis particular situation. It is more convenient for our purposes: the propertyof R(Y ) we need (see Lemma 5.5 below) becomes evident.

6. GALOIS ACTION ON GROTHENDIECK GROUP 95

Example 5.4. Let us take as L/F a quadratic extension L = F (√d) with

some d ∈ F ∗ and as T the Severi-Brauer variety of a quaternion F -algebra(a, b). Then R(T ) is the quadric determined by the quadratic form

⟨−a,−b, ab, d⟩ .

Lemma 5.5. Let L/F be a finite Galois field extension with the Galoisgroup G. Let T be an F -variety. For any τ ∈ G, the following diagram ofisomorphisms commutes

R(T )Lid⊗τ−−−→ R(T )Ly y∏

G

TL(id⊗τ) ◦ zτ−−−−−−→

∏G

TL

Proof. It is a direct consequence of Definition 5.2.

6. Galois action on Grothendieck group

In this Section, F is an arbitrary field, L/F is a field extension (e.g. aGalois extension), G is a group of automorphism of L over F (e.g. the Galoisgroup in the case where L/F is a Galois extension), Y is an F -variety.

The group G acts on the Grothendieck group K(YL) of the L-variety YL.We are interested in a condition on Y which guarantees that the action of Gon K(Y )L is trivial.

Lemma 6.1. Suppose that the group K(YL) is torsion-free and that thecokernel of the restriction map resL/F : K(Y ) → K(YL) is a torsion group.Then the action of G on K(YL) is trivial.

Proof. Take any y ∈ K(YL) and any σ ∈ G. Since Coker(resL/F ) is atorsion group, some multiple ny of y is in Im(resL/F ), therefore σ(ny) = ny.Since the group K(YL) is torsion-free, it follows that σ(y) = y.

Working with homogeneous varieties, we have the first condition of Lemma6.1 for free: the group K(Y ) is natural (with respect to extensions of the basefield F ) isomorphic to K(A), where A is a separable algebra (i.e. the directproduct of simple algebras with centers separable over F ) ([65, Introduction]).As to the second condition, it is equivalent to the condition that every simplecomponent of A is central over F . We are not interested here in a completelist of homogeneous varieties satisfying this condition. We only notice thatthe generalized Severi-Brauer varieties are included (see [69, Theorem 4.1 of§8] for the case of usual Severi-Brauer varieties and/or [50, Theorem 4.4] forthe case of generalized Severi-Brauer varieties) as well as their direct products([59, §1.8]). So that we have

Corollary 6.2. Let Y be a product of generalized Severi-Brauer vari-eties. Then the action of G on K(YL) is trivial.


Corollary 6.3. Let L/F be a finite Galois extension, G its Galois group,and Y a product of generalized Severi-Brauer varieties over F . Let us identifyR(Y )L with

∏G YL (see Definition 5.2). Then for any σ ∈ G the automor-

phism of K(R(Y )L) given by σ corresponds to the automorphism of K(∏

G YL)given by the automorphism of the product induced by the permutation zσ of thefactors, where zσ is the left translation by σ.

Proof. By Lemma 5.5, the diagram

K(R(Y )L)σ−−−→ K(R(Y )L)y y

K(∏G

YL)σ ◦ zσ−−−→ K(

∏G

YL)

commutes. By Corollary 6.2, σ over the bottom arrow is the identity.

7. Product of conics

In this Section, the ground field is denoted by l in order to adopt thenotation to the situation where the results of this Section will be applied.We fix an algebraic closure l of the field l. All algebras and varieties in thisSection are algebras and varieties over l. For any variety X, we denote by Xthe l-variety Xl. For any homogeneous variety X, we identify the ring K(X)with its image in K(X) under the restriction homomorphism which is injectivebecause the group K(X) is torsion-free ([65, Introduction]).

We also need to introduce certain terminology concerning the abelian groupswith filtration.

Let A be an abelian group. We consider only the descending filtration A(i)

(i ∈ Z) on A satisfying the condition A(0) = A.

Definition 7.1. Let A be an abelian group with filtration and a ∈ A.We define the codimension codimA a of a in A as

codimA adef= sup{i ∈ Z| a ∈ A(i)} .

Definition 7.2. Let A be an abelian group with filtration. A system ofgenerators a1, . . . , an of A is called filtering if for any i ∈ Z the term A(i) isgenerated by a subsystem of a1, . . . , an (and hence, generated by the subsystemof the elements of codimensions ≥ i). A filtering system is called a filteringbasis if the abelian group A is free and a1, . . . , an form its basis.

Clearly, to determine a filtration on A, it suffices to give a filtering systemof generators with their codimensions. Giving a filtering system of generators,we shall first write down its elements of codimension 0; then, after the “;”-sign,the elements of codimension 1; and so on.

Lemma 7.3. Let Q be a split quaternion algebra and let Y be its Severi-Brauer variety. Denote by p ∈ K(Y ) the class of a rational point. Then 1; p isa filtering basis of K(Y ). The multiplication in the ring K(Y ) is determinedby the formula p2 = 0.

7. PRODUCT OF CONICS 97

Proof. Since the quaternion algebra Q is split, the variety Y is (isomor-phic to) a projective line. The statement on the filtering basis follows now e.g.from Corollary 3.9 (note that the tautological vector bundle T corresponds tothe locally free module O(−1) and that p = 1− [T ]). Since p2 ∈ K(Y )(2) anddimY = 1, it follows that p2 = 0.

Lemma 7.4. Let Q be a quaternion division algebra and let Y be its Severi-Brauer variety. Denote by p ∈ K(Y ) the class of a rational point. Then 1; 2pis a filtering basis of K(Y ).

Proof. Since there exists a quadratic extension of L splitting Q, the trans-fer argument shows that 2p ∈ K(Y )(1). Thus K(Y ) contains the subgroup ofK(Y ) generated by 1 and 2p; the index of the latter subgroup in K(Y ) is 2.Since there is a natural (with respect to extensions of scalars) isomorphismK(Y ) ≃ K(L)⊕K(Q) ([69, Theorem 4.1 of §8]), the index of K(Y ) in K(Y )equals indQ = 2. Consequently, K(Y ) coincides with the subgroup generatedby 1 and 2p.

Now we check the statement on the filtration. We have K(Y )(1) ⊂ K(Y )∩K(Y )(1) and the intersection is evidently generated by 2p. From the otherhand, we have already shown that 2p ∈ K(Y )(1). Thus K(Y )(1) coincides withthe subgroup generated by 2p.

For two next lemmas, we fix the following notation: Q1, . . . , Qn are quater-nion algebras; for i = 1, . . . , n, let Yi be the Severi-Brauer variety of Qi andlet pri : Y1 × · · · × Yn → Yi be the projection.

Lemma 7.5. Suppose that the quaternion algebras Q1, . . . , Qn are split.Then the map

pr∗1 · · · pr∗n : K(Y1)⊗Z · · · ⊗Z K(Yn) → K(Y1 × · · · × Yn)

is an isomorphism of rings with filtration. In particular,

1; {pi}i; {pipj}i<j; . . . ; p1 · · · pnis a filtering basis of K(Y1 × · · · × Yn), where pi ∈ K(Yi) are the classes ofrational points.

Proof. Since the quaternion algebras Q1, . . . , Qn are split, the varietiesY1, . . . , Yn are (isomorphic to) projective lines.

Lemma 7.6. Suppose that the quaternion algebras Q1, . . . , Qn are such thatthe tensor product Q1 ⊗l · · · ⊗l Qn is a skewfield. Then the map

pr∗1 · · · pr∗n : K(Y1)⊗Z · · · ⊗Z K(Yn) → K(Y1 × · · · × Yn)

is an isomorphism of rings with filtration. In particular,

1; {2pi}i; {4pipj}i<j; . . . ; 2np1 · · · pnis a filtering basis of K(Y1 × · · · × Yn), where pi ∈ K(Yi) are the classes ofrational points.


Proof. The map pr∗1 · · · pr∗n : K(Y1)⊗Z · · · ⊗Z K(Yn) → K(Y1 × · · · × Yn)(which is a homomorphism of rings with filtrations) is evidently injective. Tocheck that its image, i.e. the group K(Y1) · · ·K(Yn), coincides with K(Y1 ×· · · × Yn) it suffices to check equality of the indexes (compare to the proof ofLemma 7.4). Since by [69, Theorem 4.1 of §8]

K(Y1 × · · · × Yn) ≃ K((F ×Q1)⊗l · · · ⊗l (F ×Qn)

)and for any 1 ≤ i1 < · · · < ik ≤ n the product Qi1 ⊗· · ·⊗Qik , being a divisionalgebra, has the index 2k, the index of K(Y1 × · · · × Yn) in K(Y1 × · · · × Yn)coincides with the index of K(Y1) · · ·K(Yn).

Now we check the statement on the filtrations. For any p ≥ 0, the followinginclusions are evident:(

K(Y1) · · ·K(Yn))(p) def

=∑

i1+···+in=p

K(Y1)(i1) · · ·K(Yn)

(in) ⊂

⊂ K(Y1 × · · · × Yn)(p) ⊂ K(Y1 × · · · × Yn) ∩K(Y1 × · · · × Yn)

(p) .

Since the first term coincides with the last one, we are done.

Let Ydef= SB(Q) and Y

def= SB(Qop), where Q is a quaternion algebra.

The canonical vector bundle I on Y is a right QY -module while the canonical

vector bundle I on Y is a left QY -module. Denote by E the tensor product

I ⊗QY×YI. It is a vector bundle of rank 1. Let p and p be the classes of

rational points on¯Y and ¯Y .

Lemma 7.7. In the notation introduced right above, the class of E in K(Y×Y ) equals (1− p)(1− p).

Proof. First note that 4[E ] = [I] · [I], because 4 = dimlQ. For the rest

of the proof we assume that Q is split. The varieties Y , Y are (isomorphic to)

projective spaces and [I] = 2[OY (−1)], [I] = 2[OY (−1)] ([69, §8.4]). Finally,since p is the class of a hyperplane, we have p = 1 − [OY (−1)]. Analogously,p = 1 − [OY (−1)]. So, we get the formula 4[E ] = 4(1 − p)(1 − p). Since theGrothendieck group is torsion-free, one can divide by 4.

8. Preliminary calculations I

The goal of this Section is Proposition 8.5.In this Section, the ground field is denoted by l in order to adopt the

notation to the situation where the results of this Section will be applied.We are going to treat a rather special situation which will occur in Section

10. Let∧

Q1 and∧

Q2 be quaternion division l-algebras such that the tensor

8. PRELIMINARY CALCULATIONS I 99

product∧

Qdef=

∧

Q1 ⊗l

∧

Q2 is a skew-field. For i = 1, 2, we put∨

Qidef=

∧

Qop

i ,∨

Qdef=

∧

Qop

∧

Y idef= SB(

∧

Qi),∨

Y idef= SB(

∨

Qi),∧

Tdef= SB(2,

∨

Q),∨

Tdef= SB(2,

∧

Q),

Xidef=

∧

Y i ×∨

Y i, Xdef= X1 ×X2, X

def= X ×

∧

T ×∨

T .

Let∧

I1 be the canonical vector bundle on∧

Y 1,∧

I2 the canonical vector bundle

on∧

Y 2, and∧

J the canonical vector bundle on∧

T . The tensor product∧

I1⊗∧

I2 of

the vector bundles over X is a right∧

QX-module, while∧

J is a left∧

QX-module.

Put∧

T def= (

∧

I1 ⊗∧

I2) ⊗∧QX

∧

J and denote by −∧η,

∧µ ∈ K(X) the first and the

second Chern classes of∧

T .

Let∨

I1 be the canonical vector bundle on∨

Y 1,∨


on∨

Y 2, and∨

J the canonical vector bundle on∨

T . The tensor product∨

I1⊗∨

I2 of

the vector bundles over X is a right∨

QX-module, while∨

J is a left∨

QX-module.

Put∨

T def= (

∨

I1 ⊗∨

I2) ⊗∨QX

∨

J and denote by −∨η,

∨µ ∈ K(X) the first and the

second Chern classes of∨

T .

By∧p1,

∧p2,

∨p1, and

∨p2 we denote the classes of rational points on

∧

Y 1,∧

Y 2,∨

Y 1, and∨

Y 2.

Proposition 8.1. A filtering basis of the group K(X) is given by the prod-ucts of elements of the following table such that from every column at most oneelement is taken:

1 2∧p1 2

∧p2

∧p1 +

∨p1 −

∧p1

∨p1

∧p2 +

∨p2 −

∧p2

∨p2

∧η

∨η

2∧µ,

∧η2 ∨µ,

∨η2

3∧µ

∧η

∨µ

∨η

4∧µ2 ∨

µ2

The codimension of an element in the table is the number of the line where itis placed; the codimension of an element of the filtering basis is the sum of thecodimensions of the factors.

Proof. By Corollary 2.4, the projection∧

Y 1 ×∧

Y 2 ×∧

T →∧

Y 1 ×∧

Y 2 is a

grassmanian. Thus X → X ×∨

T is a grassmanian as well (this morphism isobtained from the previous one by a base change). More precisely, it is thegrassmanian of 2-planes in a rank 4 vector bundle. Moreover, by Corollary 2.4,∧

T is the tautological vector bundle of this grassmanian. Therefore, by Corol-

lary 3.10, K(X) is a free K(X×∨

T )-module with the basis 1,∧η,

∧µ,

∧η2,

∧µ

∧η,

∧µ2and


the topological filtration on K(X) is induced by the conditions∧η ∈ K(X)(1);

∧η2,

∧µ ∈ K(X)(2);

∧η

∧µ ∈ K(X)(3);

∧µ2∈ K(X)(4).

We have reduced the problem of computation of the group K(X) with the

filtration to the similar problem for K(X ×∨

T ).

By Corollary 2.4, the projection∨

Y 1×∨

Y 2×∨

T →∨

Y 1×∨

Y 2 is a grassmanian.

Thus X ×∨

T → X is a grassmanian as well (this morphism is obtained fromthe previous one by a base change). More precisely, it is (once again) the

grassmanian of 2-planes in a rank 4 vector bundle (Corollary 2.4). Moreover,∨

Tis the tautological vector bundle of this grassmanian. Therefore, by Corollary

3.10, K(X×∨

T ) is a free K(X)-module with the basis 1,∨η,

∨µ,

∨η2,

∨µ

∨η,

∨µ2and the

topological filtration onK(X×∨

T ) is induced by the conditions∨η ∈ K(X×

∨

T )(1);∨η2,

∨µ ∈ K(X ×

∨

T )(2);∨η

∨µ ∈ K(X ×

∨

T )(3);∨µ2∈ K(X ×

∨

T )(4).

We have reduced the problem of computation of the group K(X×∨

T ) withthe filtration to the similar problem for K(X).

By Proposition 3.3, the projection∧

Y 2 ×∨

Y 2 →∧

Y 2 is a projective line

bundle. Thus X → X1 ×∧

Y 2 is a projective line bundle as well (this morphismis obtained from the previous one by a base change). Moreover, accordingto Lemma 7.7, the class of the tautological vector bundle on this grassmanian

equals (1− ∧p2)(1−

∨p2). Consequently, by Corollary 3.9, K(X) is a free K(X1×

∧

Y 2)-module with the basis 1,∧p2+

∨p2−

∧p2

∨p2; the topological filtration on K(X)

is induced by the condition∧p2 +

∨p2 −

∧p2

∨p2 ∈ K(X)(1).


filtration to the similar problem for K(X1 ×∧

Y 2).

By Proposition 3.3, the projection X1 →∧

Y 1 is a projective line bundle.

Thus X1 ×∧

Y 2 →∧

Y 1 ×∧

Y 2 is a projective line bundle as well. Moreover,according to Lemma 7.7, the class of the tautological vector bundle on this

grassmanian equals (1−∧p1)(1−

∨p1). Consequently, by Corollary 3.9,K(X1×

∧

Y 2)

is a free K(∧

T 1 ×∧

Y 2)-module with the basis 1,∧p1 +

∨p1 −

∧p1

∨p1; the topological

filtration on K(X) is induced by the condition∧p1+

∨p1−

∧p1

∨p1 ∈ K(X1×

∧

Y 2)(1).

We have reduced the problem of computation of the group K(X1 ×∧

Y 2)

with the filtration to the similar problem for K(∧

Y 1 ×∧

Y 2).

According to Lemma 7.6, the elements 1; 2∧p1, 2

∧p2; 4

∧p1

∧p2 form a filtering

basis of K(∧

Y 1 ×∧

Y 2).

A quaternion algebra has a canonical antiautomorphism (the canonicalsimplectic involution). For i = 1, 2, via this isomorphism, we can identify the

algebras∧

Qi and∨

Qi. Taking the product of the antiautomorphisms, we identify

8. PRELIMINARY CALCULATIONS I 101

the algebras∧

Q and∨

Q as well. Thus the following varieties are identified:∧

Y 1

and∨

Y 1,∧

Y 2 and∨

Y 2,∨

T and∧

T .Denote by s the automorphism of

X =∧

Y 1 ×∨

Y 1 ×∧

Y 2 ×∨

Y 2 ×∧

T ×∨

T

given by the permutation of the factors interchanging every ∧-factor with thecorresponding ∨-factor. The induced ring automorphism of K(X) will be alsodenoted by s.

Lemma 8.2. Application of s to an element of the filtering basis of K(X)given in Proposition 8.1 changes every ∧-sign to ∨-sign and vise versa.

Proof. Clearly, the vector bundles∧

T and∨

T are interchanged by s. Thus,

the following elements of K(X) are interchanged by s:∧η and

∨η;

∧µ and

∨µ.

We can also consider s as an automorphism of K(X). Clearly, the following

elements of K(X) are interchanged by s:∧p1 and

∨p1;

∧p2 and

∨p2.

Remark 8.3. Note that unfortunately s is not given by a permutation of

the basis of K(X) (although it is “almost” so): for instance, s(2∧p1) = 2

∨p1 is

not a basis element while 2∧p1 is. If x is a basis element not containing 2

∧pi

(i = 1, 2) as a factor, then s(x) is a basis element.

Denote by L the function field l(∧

T ×∨

T ) of the variety∧

T ×∨

T . We aregoing to work with the pull-back homomorphism K(X) → K(XL). First wecalculate it in terms of the basis of K(X). Since it is a homomorphism of

K(X)-algebras, it suffices to calculate the images of∧η,

∧µ,

∨η, and

∨µ.

Lemma 8.4. For the pull-back K(X) → K(XL), one has∧η 7→ 2(

∧p1 +

∧p2 −

∧p1

∧p2) ;

∨η 7→ 2(

∨p1 +

∨p2 −

∨p1

∨p2) ;

∧µ 7→ 2

∧p1

∧p2

∨µ 7→ 2

∨p1

∨p2

Proof. It suffices to check the statement over an extension of the base

field. Thus we may assume that the algebras∧

Q1 and∧

Q2 are split.

Since∧

T def= (

∧

I1⊗∧

I2)⊗∧QX

∧

J and dimF

∧

Q = 16, we have 16[∧

T ] = [∧

I1]·[∧

I2]·[∧

J ].

Applying the pull-back to the right-hand side product, we get [∧

I1] · [∧

I2] · 8because the rank of the vector bundle

∧

J equals 8. Since∧

Y 1 and∧

Y 2 are

projective lines, for i = 1, 2, we have [∧

I i] = 2∧

ξi, where∧

ξidef= [O∧

Y i(−1)].

Since the Chern classes are compatible with the pull-back, it follows that

the images of∧η and of

∧µ are respectively the first and the second Chern classes

of 2∧

ξ1∧

ξ2.

Let us compute the total Chern class ct of 2∧

ξ1∧

ξ2:

ct(2∧

ξ1∧

ξ2) =(ct(

∧

ξ1∧

ξ2))2

=(1 + (

∧

ξ1∧

ξ2 − 1)t)2.


Therefore, the first Chern class equals 2(∧

ξ1∧

ξ2 − 1) and the second Chern class

equals (∧

ξ1∧

ξ2 − 1)2. Substituting∧

ξi = 1 − ∧pi we get the statement on

∧η and

∧µ required (remember that by the definition,

∧η is the first Chern class with

minus).

The statement on∨η and

∨µ is obtained in the similar way.

Let K(X)(4) s ⊂ K(X)(4) be the subgroup of the s-invariant elements.

Let p be the class of a rational point on X. Note that p =∧p1

∨p1

∧p2

∨p2 ∈ K(X).

Proposition 8.5. 2p ∈ Im(K(X)(4) s → K(XL)).

Proof. We are going to work with the following composition:

β : K(X) → K(XL)resL/L−−−→ K(XL) � K(XL)/4 ·K(XL),

where L is an algebraic closure of L. We are going to show that β(K(X)(4)

s)=

0. Since the class of 2p in the quotientK(XL)/4·K(XL) is non-zero (see Lemma7.5), the affirmation of Proposition will then follow.

A filtering basis of the groupK(X) is given in Proposition 8.1. The elementsof this basis having codimensions ≥ 4 form a basis of the term K(X)(4) (seeDefinition 7.2). The basis of K(X)(4) contains the elements∧

ξdef= (

∧p1+

∨p1−

∧p1

∨p1)(

∧p2+

∨p2−

∧p2

∨p2)

∧µ and

∨

ξdef= (

∧p1+

∨p1−

∧p1

∨p1)(

∧p2+

∨p2−

∧p2

∨p2)

∨µ .

Denote by H the subgroup of K(X)(4) generated by all basis elements except∧

ξ and∨

ξ.

Lemma 8.6. The sum H + 2K(X)(4) lies in Ker β and is s-invariant.

Proof. Every basis element is a product of the following elements wherein the first column the codimension of the element in K(X) (see Definition7.1) is given; in the last column the image under β of the element is given (seeLemma 8.4):

(codim = 1) 2∧p1 7→ 2

∧p1

(codim = 1) 2∧p2 7→ 2

∧p2

(codim = 1)∧p1 +

∨p1 −

∧p1

∨p1 7→ ∧

p1 +∨p1 −

∧p1

∨p1

(codim = 1)∧p2 +

∨p2 −

∧p1

∨p2 7→ ∧

p2 +∨p2 −

∧p2

∨p2

(codim = 1)∧η 7→ 2(

∧p1 +

∧p2 −

∧p1

∧p2)

(codim = 1)∨η 7→ 2(

∨p1 +

∨p2 −

∨p1

∨p2)

(codim = 2)∧µ 7→ 2

∧p1

∧p2

(codim = 2)∨µ 7→ 2

∨p1

∨p2

The image of each element of the table but that of∧p1+

∨p1−

∧p1

∨p1 and

∧p2+

∨p2−

∧p2

∨p2

is divisible by 2. If we like to form a product which is a basic element witha non-zero image under β, we are allowed to take no more than one copy of∧p1 +

∨p1 −

∧p1

∨p1, no more than one copy of

∧p2 +

∨p2 −

∧p2

∨p2, and no more than

9. PRELIMINARY CALCULATIONS II 103

one other element of the table. Therefore, the only generators of K(X)(4),

which have a non-zero image with respect to β, are∧

ξ and∨

ξ. So, β(H) = 0.

Moreover, since β(∧

ξ) = 2p = β(∨

ξ), it follows that β(2∧

ξ) = 0 = β(2∨

ξ). ThusH + 2K(X)(4) ⊂ Ker β.

Since 2K(X)(4) is evidently s-invariant, it suffices to check the inclusion

s(H) ⊂ H + 2K(X)(4) .

Take a basis element of H. It is a product of the elements in the table of

Proposition 8.1. Therefore, it is either x, either (2∧p1)x, either (2

∧p2)x, either

(2∧p1)(2

∧p2)x, where x is a basis element of K(X) not containing 2

∧pi (i = 1, 2).

Since s(x) is again a basis element of H, we have no problem in the first case.

Set hidef=

∧pi +

∨pi −

∧pi

∨pi. Note that

∨pi = hi −

∧pi +

∧pihi (here the relation of

Lemma 7.3∧p2

i = 0 is used).In the second and in the third cases, we have (here again i = 1, 2):

s(2∧pix) = 2

∨pis(x) = 2his(x)− 2

∧pis(x) + 2

∧pihis(x) .

The first summand is in 2K(X)(4), the second summand is in H, the thirdsummand is in K(X)(5) ⊂ H.

Finally, in the fourth case, we have

s((2∧p1)(2

∧p2)x) = (2

∨p1)(2

∨p2)s(x) =

=(2h1 + (2

∧p1)(h1 − 1)

)(2h2 + (2

∧p2)(h2 − 1)

)s(x) ≡

≡ (2∧p1)(2

∧p2)(h1 − 1)(h2 − 1)s(x) ≡ (2

∧p1)(2

∧p2)s(x) ∈ H

where the first congruence is modulo 2K(X)(4) and the second congruence ismodulo K(X)(5) ⊂ H.

Denote by K(X)(4) the quotient K(X)(4)/(H + 2K(X)(4)). According to

Lemma 8.6, β determines a homomorphism of K(X)(4) and s determines an

automorphism of K(X)(4). To show that β((K(X)(4))s) = 0 it suffices to showthat

β((K(X)(4))s) = 0 .

The group K(X)(4) is generated by∧

ξ and∨

ξ subject to the only relations 2∧

ξ = 0,

2∨

ξ = 0. These two generators are interchanged by s. Therefore, the group

(K(X)(4))s is generated by∧

ξ +∨

ξ. Since β(∧

ξ +∨

ξ) = 2p + 2p = 0, we get therelation required.

9. Preliminary calculations II

The goal of this Section is Proposition 9.4.In this Section, the ground field is denoted by l in order to adopt the

notation to the situation where the results of this Section will be applied.


We are going to treat a special situation which will occur in Section 11.

Let∧

Q1 and∧

Q2 be quaternion division l-algebras such that the tensor product∧∧

Qdef=

∧

Q1 ⊗l

∧

Q2 is a skew-field. For i = 1, 2, we put

∨

Qidef=

∧

Qop

i ,∨∨

Qdef=

∨

Q1 ⊗∨

Q2∧∨

Qdef=

∧

Q1 ⊗∨

Q2

∨∧

Qdef=

∨

Q1 ⊗∧

Q2∧

Y idef= SB(

∧

Qi),∨

Y idef= SB(

∨

Qi),∧∧

Tdef= SB(2,

∨∨

Q),∨∨

Tdef= SB(2,

∧∧

Q),∧∨

Tdef= SB(2,

∨∧

Q),∨∧

Tdef= SB(2,

∧∨

Q),

Xidef=

∧

Y i ×∨

Y i, Xdef= X1 ×X2, X

def= X ×

∧∧

T ×∨∨

T ×∧∨

T ×∨∧

T .

Let∧

I1 be the canonical vector bundle on∧

Y 1,∧


on∧

Y 2, and∧∧

J the canonical vector bundle on∧∧

T . The tensor product∧

I1⊗∧

I2 of

the vector bundles over X is a right∧∧

QX-module, while∧∧

J is a left∧∧

QX-module.

Put∧∧

T def= (

∧

I1 ⊗∧

I2) ⊗∧∧QX

∧∧

J and denote by −∧∧η,

∧∧µ ∈ K(X) the first and the

second Chern classes of∧∧

T .

Let∨

I1 be the canonical vector bundle on∨

Y 1,∧


on∨

Y 2, and∨∧

J the canonical vector bundle on∨∧

T . The tensor product∨

I1⊗∧

I2 of

the vector bundles over X is a right∨∧

QX-module, while∨∧

J is a left∨∧

QX-module.

Put∨∧

T def= (

∨

I1 ⊗∧

I2) ⊗∨∧QX

∨∧

J and denote by −∨∧η,

∨∧µ ∈ K(X) the first and the

second Chern classes of∨∧

T .

We define the vector bundles∧∨

T and∨∨

T in the similar way:

∧∨

T def= (

∧

I1 ⊗∨

I2)⊗∧∨QX

∧∨

J∨∨

T def= (

∨

I1 ⊗∨

I2)⊗∨∨QX

∨∨

J

and denote by −∧∨η,

∧∨µ ∈ K(X) the first and the second Chern classes of

∧∨

T ; by

−∨∨η,

∨∨µ ∈ K(X) the first and the second Chern classes of

∨∨

T .

By∧p1,

∧p2,

∨p1, and

∨p2 we denote the classes of rational points on

∧

Y 1,∧

Y 2,∨

Y 1, and∨

Y 2.

Proposition 9.1. A filtering basis of the group K(X) is given by the prod-ucts of elements of the following table such that from every column at most one


element is taken:

1 2∧p1 2

∧p2

∧p1 +

∨p1 −

∧p1

∨p1

∧p2 +

∨p2 −

∧p2

∨p2

∧∧η

∧∨η

∨∧η

∨∨η

2∧∧µ,

∧∧η2 ∧∨µ,

∧∨η2 ∨∧µ,

∨∧η2 ∨∨µ,

∨∨η2

3∧∧µ

∧∧η

∧∨µ

∧∨η

∨∧µ

∨∧η

∨∨µ

∨∨η

4∧∧µ2 ∧∨

µ2 ∨∧

µ2 ∨∨

µ2

The codimension of an element in the table is the number of the line where itis placed; the codimension of an element of the filtering basis is the sum of thecodimensions of the factors.

Proof. By Corollary 2.4, the projection∨

Y 1 ×∧

Y 2 ×∨∧

T →∨

Y 1 ×∧

Y 2 is a

grassmanian. Thus X → X×∧∧

T×∨∨

T×∧∨

T is a grassmanian as well (this morphismis obtained from the previous one by a base change). More precisely, it is thegrassmanian of 2-planes in a rank 4 vector bundle. Moreover, by Corollary

2.4,∨∧

T is the tautological vector bundle of this grassmanian. Therefore, by

Corollary 3.10, K(X) is a free K(X ×∧∧

T ×∨∨

T ×∧∨

T )-module with the basis

1,∨∧η,

∨∧µ,

∨∧η2,∨∧µ

∨∧η,

∨∧µ2and the topological filtration on K(X) is induced by the

conditions∨∧η ∈ K(X)(1);

∨∧η2,∨∧µ ∈ K(X)(2);

∨∧η

∨∧µ ∈ K(X)(3);

∨∧µ2∈ K(X)(4).


filtration to the similar problem for K(X ×∧∧

T ×∨∨

T ×∧∨

T ).

By Corollary 2.4, the projection∧

Y 1×∨

Y 2×∧∨

T →∧

Y 1×∨

Y 2 is a grassmanian.

Thus X×∧∧

T ×∨∨

T ×∧∨

T → X×∧∧

T ×∨∨

T is a grassmanian as well. More precisely, it

is the grassmanian of 2-planes in a rank 4 vector bundle. Moreover,∧∨

T is thetautological vector bundle of this grassmanian (Corollary 2.4). Therefore, by

Corollary 3.10, K(X ×∧∧

T ×∨∨

T ×∧∨

T ) is a free K(X ×∧∧

T ×∨∨

T )-module with the

basis 1,∧∨η,

∧∨µ,

∧∨η2,∧∨µ

∧∨η,

∧∨µ2and the topological filtration on K(X ×

∧∧

T ×∨∨

T ×∧∨

T ) is

induced by the conditions∧∨η ∈ K(1);

∧∨η2,∧∨µ ∈ K(2);

∧∨η

∧∨µ ∈ K(3);

∧∨µ2∈ K(4).

We have reduced the problem of computation of the groupK(X×∧∧

T×∨∨

T×∧∨

T )

with the filtration to the similar problem for K(X ×∧∧

T ×∨∨

T ). A filtering basisfor the latter group is given in Proposition 8.1 (note that the notation there is

slightly different: namely,∧η =

∧∧η ,

∨η =

∨∨η ,

∧µ =

∧∧µ, and

∨µ =

∨∨µ.

A quaternion algebra has a canonical antiautomorphism (the simplecticinvolution). For i = 1, 2, via this isomorphism, we can identify the algebras∧

Qi and∨

Qi. Taking the product of the antiautomorphisms, we identify∧∧

Q with∨∨

Q and∧∨

Q with∨∧

Q. Thus the following varieties are identified:∧

Y 1 and∨

Y 1,∧

Y 2

and∨

Y 2,∨∨

T and∧∧

T ,∨∧

T and∧∨

T .


Denote by s1 the automorphism of

X =∧

Y 1 ×∨

Y 1 ×∧

Y 2 ×∨

Y 2 ×∧∧

T ×∨∨

T ×∧∨

T ×∨∧

T

given by the permutation of the factors interchanging∧

Y 1 with∨

Y 1,∧∧

T with∨∧

T ,

and∧∨

T with∨∨

T (and leaving∧

Y 2 and∨

Y 2 untouched).Denote by s2 the automorphism of X given by the permutation of the

factors interchanging∧

Y 2 with∨

Y 2,∧∧

T with∧∨

T , and∨∧

T with∨∨

T (and leaving∧

Y 1

and∨

Y 1 untouched).The induced ring automorphisms of K(X) will be also denoted by s1 and

s2.

Lemma 9.2. Application of s1 to an element of the filtering basis of K(X)given in Proposition 9.1 changes every “first” ∧-sign (i.e. a ∧-sign placed overp1 or a ∧-sign placed on the first place over η or over µ) to the ∨-sign and viseversa: every “first” ∨-sign to the ∧-sign; the “second” ∧-and-∨-signs are leftuntouched. Application of s2 to an element of the filtering basis changes every“second” ∧-sign to the ∨-sign and vise versa; the “first” ∧-and-∨-signs are leftuntouched.

Proof. We prove only the statement on s1.

Clearly, s1 interchanges∧∧

T with∨∧

T and∧∨

T with∨∨

T . Consequently, the fol-

lowing elements of K(X) are interchanged by s1:∧∧η and

∨∧η ;

∧∨η and

∨∨η ;

∧∧µ and

∨∧µ;

∧∨µ and

∨∨µ.

We can also consider s1 as an automorphism ofK(X). Clearly, the elements∧p1 and

∨p1 of K(X) are interchanged by s while the elements

∧p2 and

∨p2 are left

untouched.

Denote by L the function field of the variety∧∧

T×∨∨

T×∧∨

T×∨∧

T . We are going towork with the pull-back homomorphism K(X) → K(XL). First we calculate itin terms of the basis of K(X). Since it is a homomorphism of K(X)-algebras,

it suffices to calculate the images of∧∧η ,

∨∨η ,

∧∨η ,

∨∧η ,

∧∧µ,

∨∨µ,

∧∨µ, and

∨∧µ.

Lemma 9.3. For the pull-back K(X) → K(XL), one has∧∧η 7→ 2(

∧p1 +

∧p2 −

∧p1

∧p2) ;

∨∨η 7→ 2(

∨p1 +

∨p2 −

∨p1

∨p2) ;

∧∨η 7→ 2(

∧p1 +

∨p2 −

∧p1

∨p2) ;

∨∧η 7→ 2(

∨p1 +

∧p2 −

∨p1

∧p2) ;

∧∧µ 7→ 2

∧p1

∧p2

∨∨µ 7→ 2

∨p1

∨p2

∧∨µ 7→ 2

∧p1

∨p2

∨∧µ 7→ 2

∨p1

∧p2

Proof. See the proof of Lemma 8.4.

Denote by G the subgroup of AutK(X) generated by s1 and s2. Let

K(X)(3)G ⊂ K(X)(3)

be the subgroup of the G-invariant elements.

Let p be the class of a rational point on X. Note that p =∧p1

∨p1

∧p2

∨p2 ∈ K(X).


Proposition 9.4. 2p ∈ Im(K(X)(3)G → K(XL)).

Proof. We are going to work with the following composition:

β : K(X) → K(XL)resL/L−−−→ K(XL) � K(XL)/4 ·K(XL),

where L is an algebraic closure of L. We are going to show that the residueclass of 2p is not in β

((K(X)(3))G

).

We introduce some additional notation for elements of K(X) in order toavoid repetitions of long expressions. For i = 1, 2, we put

pidef=

∧pi

∨pi, hi

def=

∧pi +

∨pi − pi

(although it is not essential for the consequent, we remark that: pi is the classof a rational point on Xi; hi is the class of a hyperplane section if we identifyXi with the quadric hypersurface in the 3-dimensional projective space via the

Segre imbedding). Note that p = p1p2. We also put hdef= h1h2.

A filtering basis of the groupK(X) is given in Proposition 9.1. The elementsof this basis having codimensions ≥ 3 form a basis of the term K(X)(3) (seeDefinition 7.2). Denote by H the subgroup of K(X)(3) generated by all basiselements except the following ones:

2∧pih ,

∗ηh ,

∗µh ,

∗µhi

where i = 1, 2 and ∗ =∧∧,∧∨,∨∧,∨∨.

Lemma 9.5. The sum H + 2K(X)(3) lies in Ker β and is G-invariant.

Proof. Every basis element is a product of the following elements wherein the first column the codimension of the element in K(X) (see Definition7.1) is given; in the last column the image under β of the element is given (seeLemma 8.4):

(codim = 1) 2∧p1 7→ 2

∧p1

(codim = 1) 2∧p2 7→ 2

∧p2

(codim = 1) h1 7→ h1(codim = 1) h2 7→ h2(codim = 1)

∧∧η 7→ 2(

∧p1 +

∧p2 −

∧p1

∧p2)

(codim = 1)∨∨η 7→ 2(

∨p1 +

∨p2 −

∨p1

∨p2)

(codim = 1)∧∨η 7→ 2(

∧p1 +

∨p2 −

∧p1

∨p2)

(codim = 1)∨∧η 7→ 2(

∨p1 +

∧p2 −

∨p1

∧p2)

(codim = 2)∧∧µ 7→ 2

∧p1

∧p2

(codim = 2)∨∨µ 7→ 2

∨p1

∨p2

(codim = 2)∧∨µ 7→ 2

∧p1

∨p2

(codim = 2)∨∧µ 7→ 2

∨p1

∧p2

The image of each element of the table but that of h1 and h2 is divisibleby 2. Easily seen, the basis elements of K(X)(3), which have a non-zero imagewith respect to β, are precisely the basic elements excepted in the definition


of H. In particular, β(H) = 0. Moreover, since the image under β of anyexcepted basis element is divisible by 2, it follows that H +2K(X)(3) ⊂ Ker β.

Now we are going to check the G-invariance. We shall check only theaffirmation on s1 (the affirmation on s2 is checked in the similar way). Since2K(X)(3) is evidently s1-invariant, it suffices to check the inclusion

s1(H) ⊂ H + 2K(X)(3) .

Note that the elements 2∧p1, h1, h2, and h are s1-invariant.

Take a basis element of H. It is a product of the elements in the table

of Proposition 8.1. Therefore, it is either x, either (2∧p1)x, where x is a basis

element of K(X) not containing 2∧p1 as a factor. Since s1(x) is again a basis

element of H, we have no problem in the first case.In the second case, we have x ∈ K(X)(2) and

s1(2∧p1x) = 2

∨p1s1(x) = 2h1s1(x)− 2

∧p1s1(x) + 2

∧p1h1s1(x) .

The first summand is in 2K(X)(3) and the second summand is in H. We aregoing to show that the third summand is in H.

First suppose that x contains h1 as a factor. Since h1 is s1-invariant, s1(x)

contains h1 as well. Since h21 = 2∧p1

∨p1 and (

∧p1)

2 = 0 (Lemma 7.3), we have

2∧p1h1s1(x) = 0 in this case. So, consider the case where x does not contain h1as a factor. Since x also does not contain 2

∧p1, s1(x) contains neither h1 nor

2∧p1 and so, the product 2

∧p1h1s1(x) is a basis element of K(X)(4). If it is not in

H, then it should be an excepted basis element. However the excepted basis

element of the codimension 4 are h∗µ, ∗ =∧∧,∧∨,∨∧,∨∨, and 2

∧p1h1s1(x) is not of

that kind. Thus 2∧p1h1s1(x) ∈ H.

Denote by K(X)(3) the quotient K(X)(3)/(H + 2K(X)(3)). According to

Lemma 9.5, β determines a homomorphism of K(X)(3) and the group G acts

on K(X)(3). To show that β(K(X)(3)G) ∋ 2p, it suffices to show that

β((K(X)(3))G

)∋ 2p .

The group K(X)(3) is a vector Z/2-space with the basis given by the classesof the excepted basis elements:

2∧pih ,

∗ηh ,

∗µh ,

∗µhi .

It is easy to calculate their images under β:

β(2∧p1h) = 2p1h2 ; β(2

∧p2h) = 2p2h1

β(∗ηh) = 2p1h2 + 2p2h1 + 2p ; β(

∗µh) = 2p ;(†)

β(•◦µh1) = 2p1

◦p2 ; β(

•◦µh2) = 2

•p1p2 ;

where •, ◦ ∈ {∨, ∧}.

10. FIRST BASIC CONSTRUCTION 109

Since for i = 1, 2

si(2∧pih) = 2

∨pis(h) = 2hih− 2

∧pih+ 2

∧pihih =

= 2hih− 2∧pih ≡ 2

∧pih (mod 2K(X)(3)) ,

and s3−i(2∧pih) = 2

∧pih, the residue classes in K(X)(3) of 2

∧p1h and 2

∧p2h are

G-invariant. The action of G on∗ηh,

∗µh, and

∗µhi is described in Lemma 9.2

(G permutes these generators). Summarizing, one sees that the G-invariant

part of K(X)(3) is generated by

2∧p1h, 2

∧p2h,

∑∗

∗ηh,

∑∗

∗µh,

∑∗

∗µh1,

∑∗

∗µh2.

Using the formulas †, one can easily compute the images of these generatorsunder β:

β(2∧p1h) = 2p1h2 ; β(2

∧p2h) = 2p2h1 ;

β

(∑∗

∗ηh

)= 4β(

∧ηh) = 0 ; β

(∑∗

∗µh

)= 4β(

∧µh) = 0 ;

β

(∑∗

∗µh1

)= 4p1

∧p2 + 4p1

∨p2 = 0 ; β

(∑∗

∗µh2

)= 4

∧p1p2 + 4

∨p1p2 = 0 .

It follows that β((K(X)(3))G

)is the subgroup ofK(XL)/4K(XL) generated

by the residue classes of 2p1h2 = 2∧p1

∨p1(

∧p2 +

∨p2 −

∧p2

∨p2) and 2p2h1 = 2

∧p2

∨p2(

∧p1 +

∨p1 −

∧p1

∨p1). Therefore 2p = 2

∧p1

∨p1

∧p2

∨p2 ∈ β

((K(X)(3))G

)(by Lemma 7.5, the

group K(XL) is free with the basis given by the products of∧p1,

∨p1,

∧p2, and

∨p2).

10. First basic construction

Let k be a field of characteristic different from 2, containing elements

a1, b1, a2, b2, d ∈ k∗

such that ldef= k(

√d) is a field and the biquaternion l-algebra(

(a1, b1)⊗k (a2, b2))l

is a skewfield.Let T be the generalized Severi-Brauer variety (see Section 2) of rank 2

right ideals in the biquaternion k-algebra (a1, b1)⊗ (a2, b2). Denote by K thefunction field of the k-variety R(T ) = Rl/k(T ) (see Definition 5.2).

Put Ldef= K(

√d). Since R(T )l ≃ Tl × Tl (see Section 5), one has

ind((a1, b1)⊗ (a2, b2)

)L= 2

by the index reduction formula [7, Theorem 3].For i = 1, 2, let qi be the quadratic form ⟨−ai,−bi, aibi, d⟩ over K.


Theorem 10.1. For any odd field extension K ′/K, the quadratic forms(q1)K′ and (q2)K′ are non-linked. In particular, the forms q1 and q2 themselvesare non-linked.

Proof. Remember that the quadratic forms q1 and q2 are in fact definedover k and denote by X1 and X2 the projective quadrics over k determined by

q1 and q2. Set Xdef= X1 ×X2. We have to show that the degree of any closed

point on the variety XK is divisible by 4.Consider the Grothendieck group K(XK) of the variety XK supplied with

the topological filtration. Let p ∈ K(XK) denote the class of a rational point.To show that degree of every closed point on X is divisible by 4, it sufficesto show that 2p ∈ K(XK)(0), where K(XK)(0) is the 0-dimensional term ofthe topological filtration on K(XK). Since dimX = 4, we have K(XK)(0) =

K(XK)(4).

The pull-back homomorphism K(X ×R(T ))(4) → K(XK)(4), given by the

flat morphism of schemes XK → X × R(T ), is surjective by Corollary 4.3.Therefore it suffices to show that 2p is not in the image of this homomorphism.

Denote by σ the non-trivial automorphism of l over k. The group K(X ×R(T ))(4) is contained in the σ-invariant part of the group K(Xl × R(T )l)

(4).Thus it suffices to show that

2p ∈ Im(K(Xl ×R(T )l)

(4)σ → K(XL)).

For this, we apply Proposition 8.5.In order to meet the conditions of Proposition 8.5, note that for i = 1, 2,

one has Xi ≃ R(Yi), where R = Rl/k and Yi is the Severi-Brauer variety ofthe quaternion k-algebra (ai, bi) (see Example 5.4).

Thus we have X × R(T ) ≃ R(Y1 × Y2 × T ). Therefore, we can identifyXl ×R(T )l with the product

Xdef=

∧

Y 1 ×∧

Y 2 ×∧

T ×∨

Y 1 ×∨

Y 2 ×∨

T

where∧

Y i,∨

Y i are two copies of (Yi)l and∧

T ,∨

T are two copies of Tl. More-over, by Corollary 6.3, the automorphism of K(Xl × R(T )l) induced by σcorresponds to the automorphism of K(X) induced by the permutation of the

factors interchanging∧

Y i with∨

Y i and∧

T with∨

T .We have met the conditions of Proposition 8.5. Applying it, we get the

affirmation required.

Corollary 10.2. For any field k0 with char k0 = 2 there exist a fieldextension K/k0 and elements a1, a2, b1, b2, d ∈ K∗ with the following properties:

• ind((a1, b1)⊗ (a2, b2))K(√d) = 2;

• for any odd field extension K ′/K, the quadratic forms

q1def= ⟨−a1,−b1, a1b1, d⟩ and q2

def= ⟨−a2,−b2, a2b2, d⟩

are not linked over K ′.

11. SECOND BASIC CONSTRUCTION 111

Proof. Put kdef= k0(a1, b1, a2, b2, d) where a1, b1, a2, b2, d are indetermi-

nates. Then ldef= k(

√d) is a field and the biquaternion l-algebra

((a1, b1) ⊗

(a2, b2))lis a skewfield. For the fieldK ⊃ k as in Theorem 10.1, all affirmations

of Corollary hold.

11. Second basic construction

Let k be a field of characteristic different from 2, containing elements

a1, b1, a2, b2, d1, d2 ∈ k∗

such that ldef= k(

√d1,

√d2) is a field and the biquaternion l-algebra(

(a1, b1)⊗k (a2, b2))l

is a skewfield.Let T be the generalized Severi-Brauer variety (see Section 2) of rank 2

right ideals in the biquaternion k-algebra (a1, b1)⊗ (a2, b2). Denote by K thefunction field of the k-variety R(T ) = Rl/k(T ) (see Definition 5.2).

Put Ldef= K(

√d1,

√d2). Since R(T )l ≃ T×4

l (see Section 5), one has

ind((a1, b1)⊗ (a2, b2)

)L= 2

by the index reduction formula [7, Theorem 3].For i = 1, 2, let qi be the quadratic form ⟨−ai,−bi, aibi, di⟩ over K.

Theorem 11.1. Denote by X1 and X2 the projective quadric over K de-termined by q1 and q2. The Chow group CH2(X1 ×X2) has a torsion.

Proof. Put Xdef= X1 × X2 and consider the Grothendieck group K(X)

of the variety X. There is an isomorphism CH2(X) ≃ K(X)(2/3) (see [81,

§9]), where K(X)(2/3)def= K(X)(2)/K(X)(3) is the 2-codimensional successive

quotient of the topological filtration on K(X) . We are going to show thatthis quotient contains a torsion.

Denote by p ∈ K(XK) the class of a rational point. As we did all the time,we identify K(X) with a subgroup of K(XK) via the restriction homomor-phism.

Lemma 11.2. 2p ∈ K(X).

Proof. For i = 1, 2, denote by Ui Swan’s vector bundle onXi ([82]). It has

a structure of right (Qi)Xi-module, whereQidef= (ai, bi)K . For the class [Ui(2)] ∈

K(Xi) of the 2 (because 2 = dimXi) times twisted Swan’s vector bundle, thereis a formula ([32, Lemma 3.6]): [Ui(2)] = 4 + 2hi + h2i , where hi is the class ofa general hyperplane section of Xi. Lifting to X, we can consider the tensorproduct U1 ⊗X U2. It has a structure of right Q1 ⊗K Q2-module. Therefore,since indQ1⊗KQ2 = 2, the class [U1(2)⊗U2(2)] = (4+2h1+h

21)(4+2h2+h

22)

is divisible by 2 in K(X). Thus the product h21h22 is divisible by 2. Since

h21h22 = 4p, we are done.


Since one can find a field extension of K of degree 4 such that the forms q1and q2 become isotropic over this extension, one has 4p ∈ K(X)(4). Therefore,if we manage to show that 2p ∈ K(X)(3), we get an element of order 2 in thequotient K(X)/K(X)(3), namely the class of 2p. Since the groups K(X)(0/1)

and K(X)(1/2) are torsion-free (see [74, Lemme 6.3, (i)] for the statement onK(X)(1/2) ≃ CH1(X)), it will be a non-trivial torsion element in K(X)(2/3).

So, the last step in the proof of Theorem is the following

Lemma 11.3. 2p ∈ K(X)(3).

Proof. Remember that the quadratic forms q1 and q2 are in fact definedover k. Let us change the notation and from now on denote by X1 and X2 the

projective quadrics over k determined by q1 and q2. Set Xdef= X1 × X2. We

have to show that 2p ∈ K(XK)(3).

The pull-back homomorphism K(X ×R(T ))(3) → K(XK)(3), given by the

flat morphism of schemes XK → X × R(T ), is surjective by Corollary 4.3.Therefore it suffices to show that 2p is not in the image of this homomorphism.

Denote by G the Galois group of the biquadratic field extension l/k. ThegroupK(X×R(T ))(3) is contained in the G-invariant part of the groupK(Xl×R(T )l)

(3). Thus it suffices to show that

2p ∈ Im(K(Xl ×R(T )l)

(3)G → K(XL)).

For this, we apply Proposition 9.4.

In order to meet the conditions of Proposition 9.4, for i = 1, 2, put lidef=

k(√di) and denote by σi the non-trivial automorphism of l over l3−i. The

group G consists of 1, σ1, σ2, σ1σ2 and is generated by σ1, σ2.Let Yi be the Severi-Brauer variety of the quaternion k-algebra (ai, bi).

One has Xi ≃ Rli/k(Yi) (see Example 5.4). Therefore, we can identify (Xi)l

with∧

Y i ×∨

Y i, where∧

Y i and∨

Y i are two copies of the variety (Yi)l; moreover,by Lemma 5.5, the automorphism of (Xi)l given by σi corresponds to the

automorphism of∧

Y i ×∨

Y i given by σi composed with the interchanging ofthe factors. The automorphism of (Xi)l given by σ3−i corresponds to the

automorphism of∧

Y i ×∨

Y i given by σ3−i.We also can identify R(T )l with

∏G Tl. Choosing the following correspon-

dence between the signs ∧∧, ∨∨, ∧∨, ∨∧ and the elements of G:

∧∧ ↔ 1 · 1 = 1∨∨ ↔ σ1σ2∧∨ ↔ 1 · σ2 = σ2∨∧ ↔ σ1 · 1 = σ1

we identify R(T )l with∧∧

T ×∨∨

T ×∧∨

T ×∨∧

T where∧∧

T ,∨∨

T ,∧∨

T ,∨∧

T are copies of Tl. Theautomorphism ofR(T )l given by σ1 corresponds under this identification to the

automorphism of∧∧

T ×∨∨

T ×∧∨

T ×∨∧

T given by σ1 composed with the interchanging

12. QUADRATIC FORMS OVER COMPLETE FIELDS 113

of∧∧

T with∨∧

T and of∨∨

T with∧∨

T . Analogously, the automorphism of R(T )l given

by σ2 corresponds to the automorphism of∧∧

T×∨∨

T×∧∨

T×∨∧

T given by σ2 composed

with the interchanging of∧∧

T with∧∨

T and of∨∨

T with∨∧

T .Summarizing and passing to the Grothendieck group of the varieties, we

get the following commutative diagram (for i = 1, 2):

K(Rl1/k(Y1)l ×Rl2/k(Y2)l ×R(T )l)σi−−−→ K(Rl1/k(Y1)l ×Rl2/k(Y2)l ×R(T )l)y y

K(X)σi ◦ si−−−→ K(X)

where X and si are as in Proposition 8.5. By Corollary 6.2, σi over the bottomarrow is the identity.

We have met the conditions of Proposition 8.5. Applying it, we get theaffirmation required.

Theorem is proved.

Corollary 11.4. Let k be a field of characteristic = 2 and a, b, u, v, d, δ ∈k∗. Suppose that d, δ, dδ /∈ k∗2 and ((a, b) ⊗ (u, v))k(

√d,√δ) is a division alge-

bra. Put ρ = ⟨−a,−b, ab, d⟩, ψ = ⟨−u,−v, uv, δ⟩. Then there exists a fieldextension K/k such that

TorsCH2(XρK ×XψK ) ≃ Z/2Z and indC0(ρK)⊗ C0(ψK) = 2 .

Proof. To come to the situation considered above, we simply put a1 = a,b1 = b, a2 = u, b2 = v, d1 = d, and d2 = δ, so that q1 = ρ and q2 = ψ.

Let K be the field extension of k constructed in the beginning of thisSection. By Theorem 11.1, the group TorsCH2(XρK × XψK ) is non-trivial.From the other hand, by Theorem 5.7 of Chapter 4, the order of this group isat most 2. Therefore TorsCH2(XρK ×XψK ) ≃ Z/2Z.

Finally, let us note that C0(ρ) ≃ (a, b)k(√d) and C0(ψ) ≃ (u, v)k(

√δ). Con-

sequently indC0(ρK)⊗ C0(ψK) = ind((a, b)⊗ (u, v)

)L= 2.

12. Quadratic forms over complete fields

In this section we need some results concerning the Witt ring over a com-plete discrete valuation field. We fix the following notation:

• (L, v) is a complete discrete valuation field.• We set OL = {x ∈ L∗ | v(x) ≥ 0}, ML = {x ∈ L | v(x) > 0}, andUL = OL −ML = {x ∈ L | v(x) = 0}.

• Residue field L is defined as OL/ML.• For any a ∈ OL we denote by a the class of a in L = OL/ML.

If a ∈ UL, we obviously have a ∈ L∗. Let π be an element of L such thatv(π) is odd. Since L∗/L∗2 = UL/U

∗2L × {1, π}, an arbitrary quadratic form

ϕ over L can be written in the form ϕ = ⟨a1, . . . , ak⟩ ⊥ π ⟨b1, . . . , bl⟩ where


a1, . . . , ak, b1, . . . , bl ∈ UL. We define quadratic L-forms d1π(ϕ) and d2π(ϕ) asfollows:

d1π(ϕ) = ⟨a1, . . . , ak⟩an , d2π(ϕ) =⟨b1, . . . , bl

⟩an.

Remark 12.1. 1) Springer’s theorem asserts that a quadratic form ϕ andan element π ∈ L∗ determine quadratic forms d1π(ϕ) and d

2π(ϕ) uniquely up to

isomorphism 1. The maps

d1π, d2π : {isometry classes of L-forms} → {isometry classes of L-forms}

give rise to group homomorphisms W (L) → W (L), which are called the firstand the second residue class and denoted by ∂1 and ∂2 (see [48, §1 of Chapter6] or [76, Definition 2.5 of Chapter 6]).

2) In the case where ϕ is anisotropic, quadratic forms ⟨a1, . . . , ak⟩ and⟨b1, . . . , bl

⟩are anisotropic as well. Thus, in this case

d1π(ϕ) = ⟨a1, . . . , ak⟩ , d2π(ϕ) =⟨b1, . . . , bl

⟩.

Lemma 12.2. Let ϕ and τ be anisotropic quadratic forms over a completediscrete valuation field (L, v). Let π be an element of L such that v(π) is odd.Suppose that τ ⊂ ϕ. Then d1π(τ) ⊂ d1π(ϕ) and d

2π(τ) ⊂ d2π(ϕ).

Proof. Let γ be such that τ ⊥ γ = ϕ. It follows from Remark 12.1 thatd1π(τ) ⊥ d1π(γ) = d1π(ϕ) and d

2π(τ) ⊥ d2π(γ) = d2π(ϕ). Thus d1π(τ) ⊂ d1π(ϕ) and

d2π(τ) ⊂ d2π(ϕ).

Lemma 12.3. Let ϕ1 and ϕ2 be anisotropic quadratic k-forms. Let K =k((t)), and let ϕ = ϕ1 ⊥ tϕ2 be a quadratic form over K. Let L/K be an oddextension. Suppose that there exists τ ∈ GP2(L) such that τ ⊂ ϕL. Then thereexists an odd extension l/k of degree ≤ [L : K] such that at least one of thefollowing conditions holds:

• there exists ρ ∈ GP2(l) such that ρ ⊂ (ϕ1)l.• there exists ρ ∈ GP2(l) such that ρ ⊂ (ϕ2)l.• quadratic forms (ϕ1)l and (ϕ)l are linked.

Moreover, we can take l = L.

Proof. Since L/K is a finite field extension, L is a complete discretevaluation field. Let v be a valuation on L, and let l = L be the residue fieldof L. We have [l : k] = [L : K] ≤ [L : K]. Since L/K is odd, [l : k] is odd too.Besides, the ramification index e(L/K) = v(t) is odd. Thus, d1t and d

2t are well

defined. Since dim τ = 4 and det τ = 1, it follows that dim d1t (τ) and dim d2t (τ)are even, dim d1t (τ) + dim d2t (τ) = 4, and det d1t (τ) det d

2t (τ) = 1. Thus one of

the following conditions holds:

1) d1t (τ) ∈ GP2(L) and d21(τ) = 0,

2) d2t (τ) ∈ GP2(L) and d1t (τ) = 0,

1In the original version of Springer’s theorem, π is an uniformizing element of L. How-ever, we can suppose that π is an arbitrary element such that v(π) is odd because thereexists a prime element πL ∈ L such that π ≡ πL in L∗/L∗2.

13. 8-DIMENSIONAL QUADRATIC FORMS ϕ ∈ I2(F ) 115

3) dim d1t (τ) = dim d2t (τ) = 2 and d1t (τ) is similar to d2t (τ).

Clearly, d1t (ϕ) = (ϕ1)l and d2t (ϕ) = (ϕ2)l. It follows from Lemma 12.2 thatd1π(τ) ⊂ d1π(ϕ) = (ϕ1)l and d

2π(τ) ⊂ d2π(ϕ) = (ϕ2)l. Thus, we are done.

13. 8-dimensional quadratic forms ϕ ∈ I2(F )

It is an important problem to find a good classification of 8-dimensionalquadratic forms ϕ ∈ I2(F ). One of important invariants of ϕ is the Schur indexof the Clifford algebra C(ϕ). Clearly, indC(ϕ) is equal to one of the integers:1, 2, 4, or 8.

If ϕ is a “generic” 8-dimensional form with detϕ = 1, then we haveindC(ϕ) = 8. This shows that we cannot say anything specific in the caseindC(ϕ) = 8. In the case indC(ϕ) = 1 we have plenty information on thestructure of ϕ. Indeed, in this case c(ϕ) = 0, and hence ϕ ∈ I3(F ). Finally,APH implies that ϕ ∈ GP3(F ). The case indC(ϕ) = 2 is well known too (seefor example [41, Example 9.12]). Namely, for a quadratic form ϕ ∈ I2(F ) thefollowing two conditions are equivalent: a) indC(ϕ) ≤ 2; b) ϕ can be writtenin the form ϕ = ⟨⟨a⟩⟩ q, where dim q = 4.

Thus, the only open case is indC(ϕ) = 4. It is very easy to give examples ofquadratic forms ϕ with indC(ϕ) ≤ 4. If ϕ = π1 ⊥ π2 where π1, π2 ∈ GP2(F ),then c(ϕ) = c(π1) + c(π2), and hence indC(ϕ) ≤ 4. This example gives rise tothe following natural

Question 13.1. Suppose that ϕ ∈ I2(F ) is an 8-dimensional quadraticform with indC(ϕ) ≤ 4. Do there necessarily exist quadratic forms π1, π2 ∈GP2(F ) such that ϕ = π1 ⊥ π2 ?

In this section we construct a counterexample for this question. We startfrom the following

Definition 13.2 (cf. [32, §7]). Let ϕ be a quadratic form over F .

1) By S(F ) we denote the set of quadratic forms over F satisfying thefollowing condition: there exists ρ ∈ GP2(F ) such that ρ ⊂ ϕ.

2) By Sodd(F ) we denote the set of quadratic forms over F satisfying thefollowing condition: there exist an odd extension L/F and ρ ∈ GP2(L)such that ρ ⊂ ϕL. In other words,

Sodd(F ) = {ϕ | there exists an odd extension L/F such that ϕL ∈ S(L)}.

Clearly, S(F ) ⊂ Sodd(F ). We do not know if there exists a field F suchthat S(F ) = Sodd(F ).

2 Our interest in the set Sodd(F ) is motivated by thefollowing

Theorem 13.3 (see [32, Theorem 7.3]). Let ϕ be a quadratic form of di-mension ≥ 3. The group TorsG1K(Xϕ) is zero or equal to Z/2Z. The group

2In [32, Remark 7.2], it is remarked that a field F and a 7-dimensional form ϕ ∈Sodd(F ) \S(F ) can be constructed. However, recently O. Izhboldin showed that the form ϕthe author had in mind is in fact in S(F ).


TorsG1K(Xϕ) is nontrivial if and only if ϕ is anisotropic, dimϕ ≥ 5, andϕ ∈ Sodd(F ).

Proposition 13.4. Let ϕ ∈ I2(K) be an anisotropic 8-dimensional qua-dratic form such that indC(ϕ) = 4. Then the following conditions are equiva-lent:

1) ϕ ∈ S(K), i.e., there exists ρ ∈ GP2(K) such that ρ ⊂ ϕ,2) there exist ρ1, ρ2 ∈ GP2(K) such that ϕ = ρ1 ⊥ ρ2,3) ϕ and q are linked, where q is an Albert form corresponding to the algebra

C(ϕ).

Proof. 1)⇒2). Let ρ′ be a complement of ρ in ϕ. We have ϕ = ρ ⊥ ρ′.Clearly det ρ′ = 1 and dim ρ′ = 4. Therefore ρ′ ∈ GP2(K).

2)⇒3). One can write ρ1, ρ2 as follows: ρ1 = k1 ⟨⟨a1, b1⟩⟩ and ρ2 =k2 ⟨⟨a2, b2⟩⟩. Then c(q) = c(ϕ) = (a1, b1) + (a2, b2). Therefore, q is similarto the form ⟨−a1,−b1, a1b1, a2, b2,−a2b2⟩. Obviously, ϕK(

√a1) and qK(

√a1) are

isotropic. Hence ϕ and q are linked.3)⇒1). Suppose that ϕ and q are linked. Then there exists s ∈ K∗ such

that ϕK(√s) and qK(

√s) are isotropic. We claim that iW (ϕK(

√s)) ≥ 2. Sup-

pose at the moment that iW (ϕK(√s)) = 1. Then (ϕK(

√s))an is an anisotropic

Albert form. Then indC(ϕK(√s)) = 4. Since c(q) = c(ϕ), we see that

indC(qK(√s)) = 4. Hence the Albert form qK(

√s) is anisotropic, a contra-

diction. Thus iW (ϕK(√s)) ≥ 2. Hence there exists a 2-dimensional form µ such

that µ ⟨⟨s⟩⟩ ⊂ ϕ. To complete the proof it is sufficient to set ρ = µ ⟨⟨s⟩⟩.

In this section we construct some new examples of quadratic forms ϕ suchthat ϕ /∈ Sodd(K) (and hence ϕ /∈ S(K)). The main tool for our constructionis the following

Lemma 13.5. 1) Let ϕ1 and ϕ2 be anisotropic k-forms such that ϕ1, ϕ2 /∈Sodd(k). Denote by ϕ the quadratic form ϕ1 ⊥ tϕ2 over k((t)). Suppose thatϕ ∈ Sodd(k((t))). Then there exists a finite odd extension l/k such that (ϕ1)land (ϕ2)l are linked.

2) Let ϕ1 and ϕ2 be anisotropic k-forms such that ϕ1, ϕ2 /∈ S(k). Letϕ = ϕ1 ⊥ tϕ2 be a quadratic form over k((t)). Suppose that ϕ ∈ S(k((t))).Then ϕ1 and ϕ2 are linked.

Proof. It is an obvious consequence of Lemma 12.3.

Corollary 13.6. Let ϕ1 and ϕ2 be 4-dimensional k-forms not belongingto GP2(k). Suppose that (ϕ1)l and (ϕ2)l are not linked for any odd exten-sion l/k. Then the quadratic form ϕ1 ⊥ tϕ2 over k((t)) does not belong toSodd(k((t))).

Theorem 13.7. There exist a field K and an 8-dimensional quadraticform ϕ ∈ I2(K) such that indC(ϕ) = 4 but ϕ /∈ Sodd(K).


Proof. Let field k, elements a1, a2, b1, b2, d ∈ k∗, and 4-dimensional qua-dratic forms q1, q2 be as in Corollary 10.2. We set K = k((t)) and

ϕ = q1 ⊥ tq2 = ⟨−a1,−b1, a1b1, d⟩ ⊥ t ⟨−a2,−b2, a2b2, d⟩ .Clearly, dimϕ = 4+ 4 = 8 and det± ϕ = 1. In W (K) we have ϕ = (⟨⟨a1, b1⟩⟩ −⟨⟨d⟩⟩) − t(⟨⟨a2, b2⟩⟩ − ⟨⟨d⟩⟩) = ⟨⟨a1, b1⟩⟩ − t ⟨⟨a2, b2⟩⟩ + ⟨⟨d, t⟩⟩. Therefore, c(ϕ) =(a1, b1) + (a2, b2) + (d, t). Applying Tignol’s theorem [83, Proposition 2.4], wesee that

indC(ϕ) = ind((a1, b1)⊗ (a2, b2)⊗ (d, t)) =

= 2 ind((a1, b1)⊗ (a2, b2))K(√d) = 2 · 2 = 4 .

It follows from Corollary 13.6 that ϕ /∈ Sodd(K).

Corollary 13.8. The answer to Question 13.1 is negative.

Corollary 13.9. There exist a field K and an 8-dimensional quadraticform ϕ ∈ I2(K) such that TorsGiK(Xϕ) = 0 for i = 4 and TorsG4K(Xϕ) =Z/2Z.

Proof. It is an obvious consequence of Theorem 13.7 and [32, Theorem8].

Theorem 13.10. Let ϕ ∈ I2(k) be an 8-dimensional quadratic form. Thenthe following conditions are equivalent:

1) indC(ϕ) ≤ 4,2) at least one of the following conditions holds:

(a) there exist π1, π2 ∈ GP2(k) such that ϕ = π1 ⊥ π2,(b) there exist a field extension l/k of degree 2 and a quadratic form

τ ∈ GP2(l) such that ϕ = sl/k(τ).

Proof. 1)⇒2). If ϕ is isotropic, we can write ϕ as a sum ϕ = q ⊥ ⟨1,−1⟩,where q is an Albert form. Writing q in the form q = s ⟨−a,−b, ab, u, v,−uv, ⟩,we have ϕ = s ⟨⟨a, b⟩⟩ ⊥ −s ⟨⟨u, v⟩⟩. Setting π1 = s ⟨⟨a, b⟩⟩ and π2 = −s ⟨⟨u, v⟩⟩,we are done. Thus we can suppose that ϕ is anisotropic.

Since indC(ϕ) ≤ 4, there exists an Albert form q such that c(q) = c(ϕ).If q is isotropic, then indC(ϕ) ≤ 2, and hence ϕ can be written in the formϕ = ⟨⟨a⟩⟩⊗⟨b1, b2, b3, b4⟩. Setting π1 = ⟨⟨a⟩⟩⊗⟨b1, b2⟩ and π2 = ⟨⟨a⟩⟩⊗⟨b3, b4⟩, wehave ϕ = π1 ⊥ π2 and π1, π2 ∈ GP2(k). Thus in the case where q is isotropic,the proof is complete.

Now, we can suppose that ϕ and q are anisotropic. Let ρ = ϕ ⊥ tq bea quadratic form over K = k((t)). Obviously, dim ρ = 14 and ρ ∈ I3(K).

It follows from [72] that there exist d ∈ K and π ∈ P3(K(√d)) such that

ρ = ϕ ⊥ tq is similar to sK(√d)/K(

√d π′). Let L = K(

√d).

Since K∗/K∗2 = k∗/k∗2 × {1, t}, it is sufficient to consider the followingtwo cases:

• d = a ∈ k∗,• d has the form at with a ∈ k∗.


First, consider the case d = a ∈ k∗. In this case we have L = l((t)) withl = k(

√a). Then an arbitrary L-form γ can be written in the form ϕ1 ⊥ tϕ2,

where ϕ1 and ϕ2 are l-forms. We have

sL/K(γ) = sL/K(ϕ1 ⊥ tϕ2) = sl/k(ϕ1) ⊥ tsl/k(ϕ2).

Applying this formula to the case γ =√dπ′, we see that ϕ ⊥ tq is similar to

sl/k(ϕ1) ⊥ tsl/k(ϕ2). Hence, one of the k-forms sl/k(ϕ1), sl/k(ϕ2) is similar toϕ and the other is similar to q. Let i be such that sl/k(ϕi) ∼ ϕ, and let j besuch that sl/k(ϕj) ∼ q. Then dimϕi = 4 and dimϕj = 3. Since sl/k(ϕi) ∼ ϕ,there exists r ∈ k∗ such that ϕ = r · sl/k(ϕi) = sl/k(rϕi). Now it is sufficient

to prove that rϕi ∈ GP2(l). Let ϕj = ϕj ⊥ ⟨det(ϕi) det(ϕj)⟩. Obviously,

ϕi ⊥ tϕj ∈ I2(L). Clearly, ϕ1 ⊥ tϕ2 is similar to ϕi ⊥ tϕj. Therefore π′ is

similar to ϕi ⊥ tϕj, and hence π is similar to ϕi ⊥ tϕj. Since π ∈ I3(l((t))),

it follows that ϕi, ϕj ∈ I2(l). Since dimϕi = 4, we have ϕi ∈ GP2(l). Thus inthe case d ∈ k∗ we are done.

Now, consider the case d = at, a ∈ k∗. In this case L = k((t))(√at) is a

complete discrete valuation field with residue field k and uniformizing element√at. Then an arbitrary L-form γ can be written in the form ϕ1 ⊥

√atϕ2,

where ϕ1 and ϕ2 are k-forms. We have

sL/K(γ) = sL/K(ϕ1 ⊥√atϕ2)

= sL/K(⟨1⟩)⊗ ϕ1 ⊥ sL/K(⟨√at ⟩)⊗ ϕ2

= ⟨1, at⟩ ⊗ ϕ1 ⊥ ⟨1,−1⟩ ⊗ ϕ2

= (ϕ1 ⊥ ⟨1,−1⟩ ⊗ ϕ2) ⊥ t · aϕ1.

Applying this formula to the case γ =√dπ′, we see that ϕ ⊥ tq is similar to

(ϕ1 ⊥ ⟨1,−1⟩ ⊗ ϕ2) ⊥ t · aϕ1. Therefore one of the forms ϕ, q is similar toϕ1 ⊥ ⟨1,−1⟩⊗ϕ2 and the other is similar to aϕ1. Since ϕ and q are anisotropic,we see that dimϕ2 = 0. Therefore dim(ϕ1 ⊥ ⟨1,−1⟩ ⊗ ϕ2) = dim aϕ1. Hencedimϕ = dim q, a contradiction.

2)⇒1). In the case where ϕ = π1 ⊥ π2 and π1, π2 ∈ GP2(k), we haveindC(ϕ) ≤ indC(π1) · indC(π2) ≤ 2 · 2 = 4. Now, suppose that there exista field extension l/k of degree 2 and a quadratic form τ ∈ GP2(l) such thatϕ = sl/k(τ). First of all, we have dimϕ = [l : k] · dim τ = 8. Since τ ∈ I2(l),it follows that ϕ = sl/k(τ) ∈ I2(k) ([76, Corollary 14.9]). Finally, we havec(ϕ) = c(sl/k(τ)) = Trl/k(c(τ)). Therefore, indC(ϕ) ≤ 4.

Remark 13.11. 1) Setting l = k × k, one can consider Condition 2(a) ofTheorem 13.10 as a degenerate case of Condition 2(b).

2) Actually, Theorem 13.10 is an easy consequence of deep Rost’s theorem[72]. Rost’s proof uses numerous results on the algebraic groups. It would beinteresting to find a direct proof of Theorem 13.10 in the framework of theoryof quadratic forms.


14. 14-dimensional quadratic forms ϕ ∈ I3(F )

In this section we discuss the problem of classification of anisotropic formsϕ ∈ I3(K). For anisotropic quadratic forms ϕ ∈ I3(K), the following resultsare known: if dimϕ < 8, then ϕ is hyperbolic; if dimϕ = 8, then ϕ is similar toa 3-fold Pfister form; there are no anisotropic 10-dimensional forms belongingto I3(K); if dimϕ = 12, then there exist a 2-dimensional quadratic form µ anda 6-dimensional Albert form q such that ϕ = µ ⊗ q. Analyzing these results,one can see that:

• all anisotropic quadratic forms ϕ ∈ I3(K) of dimension ≤ 12 belongs toS(K),

• any quadratic form ϕ ∈ I3(K) of dimension ≤ 12 can be represented as

a sum∑k

i=1 ρi with ρi ∈ GP3(K) and k ≤ 2.

Here we consider the case dimϕ = 14. It is not difficult to construct aform of dimension 14 belonging to I3(K). Let τ ′1 and τ ′2 be pure subformsof 3-fold Pfister forms τ1 and τ2. Then for any k ∈ K∗ the quadratic formϕ = k(τ ′1 ⊥ −τ ′2) has dimension 14 and belongs to I3(K). This example givesrise to the following

Question 14.1. Suppose that ϕ ∈ I3(K) is a 14-dimensional quadraticform. Do there necessarily exist quadratic forms τ1, τ2 ∈ P3(K) and k ∈ K∗

such that ϕ = k(τ ′1 ⊥ −τ ′2) ?We have the following

Proposition 14.2. Let ϕ ∈ I3(K) be an anisotropic 14-dimensional form.The following conditions are equivalent:

1) ϕ ∈ S(K), i.e., there exists ρ ∈ GP2(K) such that ρ ⊂ ϕ,2) There exist ρ1, ρ2 ∈ GP3(K) such that ϕ = ρ1 + ρ2 in W (K),3) There exist τ1, τ2 ∈ P3(K) and k ∈ K∗ such that ϕ = k(τ ′1 ⊥ −τ ′2). Here

τ ′1 and τ ′2 denote pure subforms of Pfister forms τ1, τ2,4) There exist τ1, τ2 ∈ P3(K) such that ϕ ≡ τ1 + τ2 (mod I4(K)),5) e3(ϕ) is a sum of two symbols, i.e., there exist a1, b1c1, a2, b2, c2 ∈ K∗

such that e3(ϕ) = (a1, b1, c1) + (a2, b2, c2).

Proof. 1)⇒2). Let s ∈ K∗ be such that ρF (√s) is isotropic. Since

ρ ∈ GP2(K), it follows that iW (ϕK(√s)) ≥ 2. Therefore dim(ϕK(

√s))an ≤ 10,

and hence Pfister’s theorem [68] implies that dim(ϕK(√s))an ≤ 8. Thus,

iW (ϕK(√s)) ≥ 3. Hence there exists a 3-dimensional form µ such that µ ⟨⟨s⟩⟩ ⊂

ϕ. We set ρ1 = (µ ⊥ ⟨detµ⟩) ⟨⟨s⟩⟩. Clearly, ρ1 ∈ GP3(K). Let ρ2 = (ϕ ⊥−ρ1)an. We have ϕ = ρ1 + ρ2 in W (K). It is sufficient to prove thatρ2 ∈ GP3(K). Since dimϕ = 14 > 8 = dim ρ1 and ϕ = ρ1 + ρ2, it followsthat ρ2 = 0. Since ϕ, ρ1 ∈ I3(K), it follows that ρ2 ∈ I3(K). Therefore,dim ρ2 ≥ 8. Since ρ1 and ϕ contain a common 6-dimensional form µ ⟨⟨s⟩⟩,we have dim ρ2 = dim(ϕ ⊥ −ρ1)an ≤ 14 + 8 − 2 · 6 = 10. Since ρ2 isanisotropic and ρ2 ∈ I3(K), Pfister’s theorem implies that dim ρ2 = 8. There-fore, ρ2 ∈ GP3(K).


2)⇒3). It is a particular case of [17, Lemma 3.2] (see also [9, Theorem4.5])

3)⇒4). Since k(τ ′1 ⊥ −τ ′2) ≡ τ1 + τ2 (mod I4(K)), we are done.4)⇒1). Let L = K(

√s) be a field extension such that (ρ2)L is isotropic.

We have ϕL(ρ1) ≡ (ρ1 + ρ2)L(ρ1) = 0 (mod I4(L(ρ1))). Since dimϕ = 14 < 16,APH implies that ϕL(ρ1) is hyperbolic. Hence there exists an L-form γ suchthat (ϕL)an = (ρ1)L · γ. Hence, dim(ϕL)an is divisible by 8. Since dimϕ = 14,it follows that iW (ϕL) ≥ (14 − 8)/2 = 3. Since L = K(

√s), there exists a 2-

dimensional form µ such that ⟨⟨s⟩⟩µ ⊂ ϕ. Now it is sufficient to set ρ = ⟨⟨s⟩⟩µ.4)⇐⇒5). It is an easy consequence of bijectivity of e3 : I3(K)/I4(K) →

H3(K).

Theorem 14.3. There exist a field E and a 14-dimensional quadraticform τ ∈ I3(E) such that τ /∈ Sodd(E).

Proof. Let K and ϕ ∈ I2(K) be as in Theorem 13.7. Since indC(ϕ) = 4,there exists an Albert form q such that c(ϕ) = c(q). Let E = K((t)), andlet τ = ϕ ⊥ tq be a quadratic form over E. Clearly, dimϕ = 14. We havec(τ) = c(ϕ)+ c(q) = 0. Therefore τ ∈ I3(E). To complete the proof, it sufficesto verify that τ /∈ Sodd(E)

Suppose at the moment that τ ∈ Sodd(E). By Theorem 13.7, we haveϕ /∈ Sodd(K). Since q is an anisotropic Albert form, it follows that q /∈ Sodd(K).Now, it follows from Lemma 13.5 that there exists an odd extension L/K suchthat ϕL and qL are linked. Proposition 13.4 implies that ϕL ∈ S(L). SinceL/K is an odd extension, we have ϕ ∈ Sodd(K), a contradiction.

Corollary 14.4. The answer to Question 14.1 is negative.

Corollary 14.5. There exist a field K and a 14-dimensional form ϕ ∈I3(K) such that e3(ϕ) cannot be represented as a sum of two symbols.

Remark 14.6. It was proved by D. W. Hoffmann and the first author(independently) that an arbitrary 14-dimensional quadratic form ϕ ∈ I3(K)can be written in the form τ1 + τ2 + τ3 in W (K) where τ1, τ2, τ3 ∈ GP3(K)(see for instance [21]). In particular, e3(ϕ) can be represented as a sum of 3symbols.

Remark 14.7. Let n be an even integer such that n > 14. It is notdifficult to construct a field E and a quadratic form ϕ ∈ I3(E) of dimensionn such that ϕ /∈ Sodd(E). The following example shows how to construct aquadratic form ϕ ∈ I3(E) of dimension 6n (n ≥ 4) so that ϕ /∈ Sodd(E).

Example 14.8. Let n ≥ 4, and let k0 be an arbitrary field of characteris-tic = 2. Let k = k0(X1, . . . , Xn, Y1, . . . , Yn, U1, . . . , Un, V1, . . . , Vn). For any i =1, . . . , n we setAi = (X1, Y1)⊗k(Ui, Vi) and qi = ⟨−Xi,−Yi, XiYi, Ui, Vi,−UiVi⟩.Let A = A1 ⊗k · · · ⊗k An and K = k(SB(A)). Let 1 ≤ i < j ≤ n. Bythe index reduction formula [78], we have ind(Ai ⊗k Aj)K = min(ind(Ai ⊗k

Aj), ind(Ai ⊗k Aj ⊗k A)) = min(42, 4n−2) = 16. Therefore, for any odd exten-sion L/K we have ind(Ai ⊗k Aj)L = 16. Then (qi)L and (qj)L are not linked.

15. NONSTANDARD ISOTROPY 121

Now we set E = K((t1)) . . . ((tn)) and ϕ = t1(q1)E ⊥ · · · ⊥ tn(qn)E. We havec(ϕ) = [(A1)E] + · · · + [(An)E] = [AE] = [(Ak(SB(A)))E] = 0. Hence ϕ ∈ I3(E).Applying Lemma 13.5, one can show that ϕ /∈ Sodd(E).

15. Nonstandard isotropy

Let ϕ and ψ be anisotropic quadratic forms over F . An important problemin the algebraic theory of quadratic forms is to find conditions on ϕ and ψ sothat ϕF (ψ) is isotropic. In the case where dimϕ ≤ 6 the problem was studiedby many authors: the case dimϕ ≤ 4 was studied by Schapiro in [75]; the casedimϕ = 5 was studied by D. W. Hoffmann in [15]; for 6-dimensional forms ϕthe problem was studied by D. W. Hoffmann ([16]), A. Laghribi ([44], [45]),D. Leep ([49]), A. S. Merkurjev ([54]), and in Chapters 3, 4.

In these papers the authors show that under certain conditions on ϕ and ψthe isotropy of ϕ over F (ψ) is standard in a sense. Let us recall the definitionof “standard isotropy” given in Chapter 4. 3

Definition 15.1. Let ϕ and ψ be anisotropic quadratic forms such thatϕF (ψ) is isotropic. We say that the isotropy of ϕF (ψ) is standard, if at least oneof the following conditions holds:

• ψ is similar to a subform in ϕ;• there exists a subform ϕ0 ⊂ ϕ with the following two properties:


Otherwise, we say that the isotropy is non-standard.

The main theorem of Chapter 4 asserts that in the case dimϕ ≤ 6, theisotropy ϕF (ψ) is standard except (possibly) the following case: dimϕ = 6,dimψ = 4, 1 = det± ϕ = det± ψ = 1, and indC0(ϕ) = 2 = indC0(ϕ)⊗F C0(ψ).

In this section we show that there exist a 6-dimensional quadratic formϕ and a 4-dimensional quadratic form ψ such that ϕF (ψ) is isotropic, but theisotropy is not standard. More precisely, we prove the following

Theorem 15.2. Let k be a field of characteristic = 2, and let a, b, u, v, d, δ ∈k∗. Suppose that d, δ, dδ /∈ k∗2 and ((a, b) ⊗ (u, v))k(

√d,√δ) is a division alge-

bra. Then there exist a field extension K/k and c ∈ K∗ with the followingproperties:

1) Quadratic forms ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩ and ψ = ⟨−u,−v, uv, δ⟩ areanisotropic, and ϕK(ψ) is isotropic,

2) the isotropy ϕK(ψ) is not standard.

Proof. Let ρ = ⟨−a,−b, ab, d⟩. It follows from Corollary 11.4 that thereexists a field extension K/k such that TorsCH2((Xψ)K × (Xρ)K) = Z/2Z andindC0(ψK) ⊗ C0(ρK) = 2. To complete the proof, it is sufficient to applyTheorem 9.1 of Chapter 4.

3If dimϕ ≤ 6, this definition coincides with definitions given in [23] and [30]. In thissection we consider only the case dimϕ ≤ 6.


Let ϕ be aK-form and E/F be a field extension. We recall that a quadraticform ϕ is called E-minimal [20, Definition 1.1] if the following conditions hold:

• ϕ is anisotropic,• ϕE is isotropic,• (ϕ0)E is anisotropic for any form ϕ0 ⊂ ϕ with dimϕ0 < dimϕ.

Lemma 15.3. Let ϕ be a 6-dimensional and ψ a 4-dimensional quadraticforms over K. Suppose that ϕ is anisotropic and ϕF (ψ) is isotropic. Then thefollowing conditions are equivalent:

1) the isotropy ϕK(ψ) is not standard,2) ϕ is a K(ψ)-minimal form.

Proof. 1)⇒2). Suppose at the moment that ϕ is not F (ψ)-minimal. Thenthere exists ϕ0 ⊂ ϕ with dimϕ0 < dimϕ such that (ϕ0)F (ψ) is isotropic. Theisotropy (ϕ0)F (ψ) is standard because the dimension of ϕ0 is ≤ 5. The defi-nition of standard isotropy shows that the isotropy ϕK(ψ) is standard too, acontradiction.

2)⇒1). Suppose that isotropy ϕF (ψ) is standard. Then at least one of thecases of Definition 15.1 holds. First suppose that ψ is similar to a subformof ϕ. Let ϕ0 ⊂ ϕ be such that ψ ∼ ϕ0. Clearly, (ϕ0)F (ψ) is isotropic anddimϕ0 = 4 < 6 = dimϕ. Therefore ϕ is not F (ψ)-minimal, a contradiction.Now, consider the second case in Definition 15.1, i.e., suppose that there existsa subform ϕ0 ⊂ ϕ which is a Pfister neighbor such that (ϕ0)F (ψ) is isotropic. Ifdimϕ0 < dimϕ, then ϕ is not a F (ψ)-minimal, and we have a contradiction.Now, let dimϕ0 = dimϕ = 6. Then ϕ = ϕ0 is a 6-dimension Pfister neighbor.Since ϕF (ψ) is isotropic, it follows that an arbitrary 5-dimensional subform ofϕ is isotropic over F (ψ). Hence, ϕ is not F (ψ)-minimal, a contradiction.

Corollary 15.4. Let ψ be an anisotropic 4-dimensional quadratic formover k with det± ψ = 1. Then there exist a field K and a 6-dimensional formϕ over K such that ϕ is a K(ψ)-minimal form.

Proof. Replacing ψ by a similar form, we can suppose that ψ has theform ⟨−u,−v, uv, δ⟩. Replacing k by a field of rational functions k(a, b, d), wecan suppose that there exist a, b, d ∈ k∗ such that d, δ, dδ /∈ k∗ and ((a, b) ⊗(u, v))k(

√d,√δ) is a division algebra. Let K/k and c ∈ K∗ be as in Theorem

15.2. Let ϕ = ⟨⟨a, b⟩⟩ ⊥ −c ⟨⟨d⟩⟩. Theorem 15.2 implies that ϕK(ψ) is isotropic,but isotropy is not standard. Lemma 15.3 shows that ϕ is a K(ψ)-minimalform.

CHAPTER 6

On the group H3(F (ψ,D)/F )

Let F be a field of characteristic different from 2, ψ a quadratic F -form ofdimension ≥ 5, and D a central simple F -algebra of index 8 and exponent2. We denote by F (ψ,D) the function field of the product Xψ × XD, whereXψ is the projective quadric determined by ψ and where XD is the Severi-Brauer variety determined by D. We compute the relative Galois cohomologygroup H3(F (ψ,D)/F ) (with the coefficients Z/2) under the assumption thatthe index of D goes down when extending the scalars to F (ψ). Using this, wegive new, very short proofs for the following results:

– Theorem 7.1, originally proved by Laghribi in [47] and– Theorem 7.2, originally proved by Esnault, Kahn, Levine, and Viehwegin [10].

We also generalize the computation of H3(F (ψ,D)/F ) to the case of arbitraryindD.


0. Introduction

Let ψ be a quadratic form and D be an exponent 2 central simple algebraover a field F (always assumed to be of characteristic not 2). Let Xψ be theprojective quadric determined by ψ, XD the Severi-Brauer variety determinedby D, and F (ψ,D) the function field of the product Xψ ×XD.

A computation of the relative Galois cohomology group

H3(F (ψ,D)/F )def= ker

(H3(F,Z/2) → H3(F (ψ,D),Z/2)

)played a crucial role in obtaining the results of Chapters 3 and 7 concerningthe problem of isotropy of quadratic forms over the function fields of quadrics.

The group H3(F (ψ,D)/F ) is closely related to the Chow group CH2(Xψ×XD) of 2-codimensional cycles on the product Xψ × XD. The main result ofthis chapter is Theorem 8.1, where the both groups are computed assumingdimψ ≥ 5 and the index of D goes down when extending the scalars to thefunction field of ψ.

The essential part of the proof is Theorem 6.9 dealing with the case whereD is a division algebra of degree 8. This Theorem has two applications inthe theory of quadratic forms: a new shorter proof of Theorem 7.1, originallyproved by Laghribi ([47, Theoreme 1]), and a new, shorter, and more elemen-tary proof of Theorem 7.2, originally proved by Esnault, Kahn, Levine, andViehweg ([10, Corollary 9.2]).

123

124 6. ON THE GROUP H3(F (ψ,D)/F )

An important role in the proof of Theorem 6.9 plays the formula of Propo-sition 4.4, which is in fact applicable to a wide class of algebraic varieties.


1.1. Quadratic forms. Mainly, we use notation of [48] and [76]. How-ever there is certain slight difference: we denote by ⟨⟨a1, . . . , an⟩⟩ the n-foldPfister form

⟨1,−a1⟩ ⊗ · · · ⊗ ⟨1,−an⟩and by Pn(F ) the set of all n-fold Pfister forms; GPn(F ) is the set of formssimilar to a form from Pn(F ).

We recall that a quadratic form ψ is called a (Pfister) neighbor (of a Pfisterform π), if it is similar to a subform in π and dimϕ > 1

2dimπ. Two quadratic

forms ϕ and ϕ∗ are half-neighbors, if dimϕ = dimϕ∗ and there exists s ∈ F ∗

such that the sum ϕ⊥sϕ∗ is similar to a Pfister form.For a quadratic form ϕ of dimension ≥ 3, we denote by Xϕ the projective

variety given by the equation ϕ = 0 and we set F (ϕ) = F (Xϕ).

1.2. Generic splitting tower. Let γ be a non-hyperbolic quadratic form

over F . Put F0def= F and γ0

def= γan. For i ≥ 1 let Fi

def= Fi−1(γi−1) and

γidef= ((γi−1)Fi)an. The smallest h such that dim γh ≤ 1 is called the height of

γ. The sequence F0, F1, . . . , Fh is called the generic splitting tower of γ ([40]).We need some properties of the fields Fs:

Lemma 1.2.1 ([41]). Let M/F be a field extension such that dim(γM)an =dim γs. Then the field extension MFs/M is purely transcendental.

The following lemma is a consequence of the index reduction formula [55].

Lemma 1.2.2 (see [22, Theorem 1.6] or [19, Proposition 2.1]). Let ϕ be aquadratic form from I2(F ) with indC(ϕ) ≥ 2r > 1. Then there is s with0 ≤ s ≤ h(ϕ) such that dimϕs = 2r + 2 and indC(ϕs) = 2r.

Corollary 1.2.3. Let ϕ ∈ I2(F ) be a quadratic form with ind(C(ϕ)) ≥ 8.Then there is s (0 ≤ s ≤ h(ϕ)) such that dimϕs = 8 and indC(ϕs) = 8.

1.3. Central simple algebras. We are working with finite-dimensionalassociative algebras over a field. Let D be a central simple F -algebra. Wedenote by XD the Severi-Brauer variety of D and by F (D) the function fieldF (XD).

For another central simple F -algebra D′ and for a quadratic F -form ψ of

dimension ≥ 3, we set F (D′, D)def= F (XD′×XD) and F (ψ,D)

def= F (Xψ×XD).

1.4. Galois cohomology. By H∗(F ) we denote the graded ring of Galoiscohomology

H∗(F,Z/2Z) = H∗(Gal(Fsep/F ),Z/2).

For any field extension L/F , we set H∗(L/F )def= ker(H∗(F ) → H∗(L)).

2. THE GROUP TorsG∗K(X) 125

We use the standard canonical isomorphisms H0(F ) = Z/2Z, H1(F ) =F ∗/F ∗2, and H2(F ) = Br2(F ).

We also work with the cohomology groups Hn(F,Q/Z(i)), i = 0, 1, 2, de-fined by B. Kahn (see [29]). For any field extension L/F , we set

H∗(L/F,Q/Z(i)) def= ker

(H∗(F,Q/Z(i)) → H∗(L,Q/Z(i))

).

For n = 1, 2, 3, the group Hn(F ) is naturally identified with

Tors2Hn(F,Q/Z(n− 1)) .

1.5. K-theory and Chow groups. We are mainly working with smoothalgebraic varieties over a field, although the smoothness assumption is notalways essential.

Let X be a smooth algebraic F -variety. The Grothendieck ring of X isdenoted by K(X). This ring is supplied with the filtration “by codimensionof support” (which respects the multiplication); the adjoint graded ring isdenoted by G∗K(X). There is a canonical surjective homomorphism of thegraded Chow ring CH∗(X) onto G∗K(X); its kernel consists only of torsionelements and is trivial in the 0-th, 1-st and 2-nd graded components ([81, §9]).In particular we have the following

Lemma 1.5.1. The homomorphism CHi(X) → GiK(X) is bijective if atleast one of the following conditions holds:

• i = 0, 1, or 2,• CHi(X) is torsion-free.

Let X be a variety over F and E/F be a field extension. We denote byiE/F the restriction homomorphism K(X) → K(XE). The same notation weuse for the restriction homomorphisms CH∗(X) → CH∗(XE) and G

∗K(X) →G∗K(XE). We fix a separable closure F of the ground field F and denote by Xthe varietyXF . The image of the restriction homomorphism iF /F : G∗K(X) →G∗K(X) is denoted by G∗K(X). The image of the restriction homomorphismiF /F : CH∗(X) → CH∗(X) is denoted by CH

∗(X).

For a projective homogeneous variety X, we identify K(X) with a subringof K(X) via the restriction homomorphism iF /F : K(X) → K(X) which isinjective by [65].

We denote by |S| the order of a set S (if S is infinite we set |S| = ∞).

2. The group TorsG∗K(X)

Lemma 2.1. Let X be a variety over F and E/F be a field extension suchthat the homomorphism iE/F : K(X) → K(XE) is injective and the factorgroup K(XE)/iE/F (K(X)) is finite. Then

| ker(G∗K(X) → G∗K(XE)| =|G∗K(XE)/iE/F (G

∗K(X))||K(XE)/iE/F (K(X))|

Proof. The proof is the same as the proof of [33, Proposition 2].


Lemma 2.2. Let X be a variety, i be an integer, and E/F be a field ex-tension such that the group GiK(XE) is torsion-free. Then

ker(GiK(X) → GiK(XE)) = TorsGiK(X) .

Proof. Since GiK(XE) is torsion-free, one has

ker(GiK(X) → GiK(XE)) ⊃ TorsGiK(X) .

On the other hand, transfer (and specialization) arguments show that

ker(GiK(X) → GiK(XE)) ⊂ TorsGiK(X) .

Lemma 2.3. Let X be a smooth variety, i be an integer, and E/F be afield extension such that the group CHi(XE) is torsion-free. Then

• CHi(XE) ≃ GiK(XE) (and hence the group GiK(XE) is torsion-free),• CHi(XE)/iE/F (CH

i(X)) ≃ GiK(XE)/iE/F (GiK(X)).

Proof. The first assertion is contained in Lemma 1.5.1. The homomor-phism CHi(XE) → GiK(XE) induces a homomorphism

CHi(XE)/iE/F (CHi(X)) → GiK(XE)/iE/F (G

iK(X))

which is bijective since CHi(XE) → GiK(XE) is bijective and CHi(X) →GiK(X) is surjective.

Proposition 2.4. Suppose that a smooth F -variety X and a field exten-sion E/F satisfy the following three conditions:

• the homomorphism iE/F : K(X) → K(XE) is injective,• the factorgroup K(XE)/iE/F (K(X)) is finite,• the group CH∗(XE) is torsion-free.

Then

|TorsG∗K(X)| =|G∗K(XE)/iE/F (G

∗K(X))||K(XE)/iE/F (K(X))|

=|CH∗(XE)/iE/F (CH

∗K(X))||K(XE)/iE/F (K(X))|

Proof. It is an obvious consequence of Lemmas 2.1, 2.2, and 2.3.

3. Auxiliary lemmas

For an abelian group A we use notation rk(A) = dimQ(A⊗Z Q).

Lemma 3.1. Let A0 ⊂ A, B0 ⊂ B be free abelian groups such that rkA0 =rkA = rA, rkB0 = rkB = rB. Then∣∣∣∣ A⊗Z B

A0 ⊗Z B0

∣∣∣∣ = ∣∣∣∣ AA0

∣∣∣∣rB ·∣∣∣∣ BB0

∣∣∣∣rA .Proof. One has

(A⊗B)/(A0 ⊗B) ≃ (A/A0)⊗B ≃ (A/A0)⊗ ZrB ≃ (A/A0)rB ,

(A0 ⊗B)/(A0 ⊗B0) ≃ A0 ⊗ (B/B0) ≃ ZrA ⊗ (B/B0) ≃ (B/B0)rA .

3. AUXILIARY LEMMAS 127

Therefore, ∣∣∣∣ A⊗B

A0 ⊗B0

∣∣∣∣ = ∣∣∣∣ A⊗B

A0 ⊗B

∣∣∣∣ · ∣∣∣∣ A0 ⊗B

A0 ⊗B0

∣∣∣∣ = ∣∣∣∣ AA0

∣∣∣∣rB ·∣∣∣∣ BB0

∣∣∣∣rA .The following lemma is well known.

Lemma 3.2. Let A be an abelian group with a finite filtration A = F0A ⊃F1A ⊃ · · · ⊃ FkA = 0. Let B be a subgroup of A with the filtration FpB =B∩FpA. Let G∗A =

⊕p≥0 FpA/Fp+1A and G∗B =

⊕p≥0 FpB/Fp+1B. Then

• |A/B| = |G∗A/G∗B|,• if A is a finitely generated group then rkG∗A = rkA.

In the following lemma the term “ring” means a commutative ring withunit.

Lemma 3.3. Let A and B be rings whose additive groups are finitely gen-erated abelian groups. Let I be a nilpotent ideal of A such that A/I ≃ Z. LetR be a subring of A ⊗Z B and AR be a subring of A such that AR ⊗ 1 ⊂ R.Then the following inequality holds∣∣∣∣A⊗Z B

R

∣∣∣∣ ≤ ∣∣∣∣ AAR∣∣∣∣rB ·

∣∣∣∣ A⊗Z B

R + (I ⊗Z B)

∣∣∣∣rAwhere rA = rkA and rB = rkB.

Proof. Let us denote by BR the image of R under the following compo-sition A⊗B → (A/I)⊗B ≃ Z⊗B ≃ B. Obviously,∣∣∣∣ A⊗Z B

R + (I ⊗Z B)

∣∣∣∣ = ∣∣∣∣ BBR

∣∣∣∣ .For any p ≥ 0 we set FpA = {a ∈ A | ∃m ∈ N such that ma ∈ Ip}. Clearly,

Tors(A/FpA) = 0, and so A/Fp is a free abelian group. Therefore all factorgroups FpA/Fp+1A (p = 0, 1, . . . ) are free abelian. Since A/I ≃ Z, it followsthat F1A = I. Thus A/F1A ≃ Z. Since I is a nilpotent ideal of A, thereexists k such that Ik = 0. Then FkA = 0. Thus the filtration A = F0A ⊃F1A ⊃ F2A ⊃ . . . is finite and results of Lemma 3.2 can be applied.

Let FpARdef= R ∩ FpA, Fp(A ⊗ B)

def= im(FpA ⊗ B → A ⊗ B), and

FpRdef= R ∩ Fp(A ⊗ B). If K is one of the rings A, AR, A ⊗ B, or R,

we set GpKdef= FpK/Fp+1K and G∗K

def=⊕

p≥0FpK/Fp+1K. Obviously,

FpK · F qK ⊂ Fp+qK for all p and q. Therefore, K = F0K ⊃ F1K ⊃ · · · ⊃FpK ⊃ . . . is a ring filtration. Hence, the adjoint graded group G∗K has agraded ring structure. Since the additive group of B is free, we have a naturalring isomorphism G∗A⊗B ≃ G∗(A⊗B).

Since AR ⊗ 1 ⊂ R, we have G∗AR ⊗ 1 ⊂ G∗R. Clearly G0(A ⊗ B) =(A/I) ⊗ B, and G0R coincides with the image of the composition R → A ⊗B → (A/I) ⊗ B. By definition of BR, one has G0R = 1G∗A ⊗ BR (here


1G∗A denotes the unit of the ring G∗A). Therefore 1G∗A ⊗ BR ⊂ G∗R. SinceG∗AR⊗1 ⊂ G∗R, 1G∗A⊗BR ⊂ G∗R, and G∗R is a subring of G∗A⊗B, we haveG∗AR⊗BR ⊂ G∗R. Therefore |G∗(A⊗B)/G∗R| ≤ |(G∗A⊗B)/(G∗AR⊗BR)|.Applying Lemmas 3.1 and 3.2, we have∣∣∣∣A⊗B

R

∣∣∣∣ = ∣∣∣∣G∗(A⊗B)

G∗R

∣∣∣∣ ≤ ∣∣∣∣ G∗A⊗B

G∗AR ⊗BR

∣∣∣∣ = ∣∣∣∣ G∗A

G∗AR

∣∣∣∣rB ·∣∣∣∣ BBR

∣∣∣∣rA =

=

∣∣∣∣ AAR∣∣∣∣rB ·

∣∣∣∣ BBR

∣∣∣∣rA =

∣∣∣∣ AAR∣∣∣∣rB ·

∣∣∣∣ A⊗Z B

R + (I ⊗Z B)

∣∣∣∣rA .

4. On the group CH∗(X × Y )

Let X be a smooth variety. We denote by FpCH∗(X) the group⊕i≥p

CHi(X) .

Let Y be one more smooth variety. For a subgroup A of CH∗(X) and asubgroup B of CH∗(Y ), we denote by A � B the image of the compositionA⊗B → CH∗(X)⊗ CH∗(Y ) → CH∗(X × Y ).

The following assertion is evident (see also [39, §3] and Proposition 4.1 ofChapter 5).

Lemma 4.1. Let X and Y be smooth varieties over F . Then

• the natural homomorphism CH∗(X × Y ) → CH∗(YF (X)) is surjective,• the kernel of the homomorphism CH∗(X × Y ) → CH∗(YF (X)) containsthe group F1CH∗(X)� CH∗(Y ).

Corollary 4.2. If the natural homomorphism

CH∗(X)⊗ CH∗(Y ) → CH∗(X × Y )

is bijective and CH∗(Y ) is torsion-free then the homomorphism CH∗(X×Y ) →CH∗(YF (X)) induces an isomorphism

CH∗(X × Y )

F1CH∗(X)� CH∗(Y )→ CH∗(YF (X)).

Proof. Since

CH∗(X)⊗ CH∗(Y ) ≃ CH∗(X × Y ) and CH∗(X)/F1CH∗(X) ≃ CH0(X),

the factor group CH∗(X × Y )/(F1CH∗(X)� CH∗(Y )) is isomorphic to

CH0(X)⊗Z CH∗(Y ) ≃ Z⊗Z CH∗(Y ) ≃ CH∗(Y ) .

Thus, it is sufficient to prove that the homomorphism CH∗(Y ) → CH∗(YF (X))is injective. It is obvious since CH∗(Y ) is torsion-free.

5. THE GROUP TorsCH2(Xψ ×XD) 129

Corollary 4.3. Let X and Y be smooth varieties and E/F be a field ex-tension such that the natural homomorphism CH∗(XE)⊗CH∗(YE) → CH∗(XE×YE) is bijective and CH∗(YE) is torsion-free. Then there exists an isomorphism

CH∗(XE × YE)

iE/F (CH∗(X × Y )) + F1CH∗(XE)� CH∗(YE)

≃CH∗(YE(X))

iE(X)/F (X)(CH∗(YF (X)))

Proof. Obvious in view of Corollary 4.2.

Proposition 4.4. Let X and Y be smooth varieties over F and E/F bea field extension such that the following conditions hold

• CH∗(XE) is a free abelian group of rank rX ,• CH∗(YE) is a free abelian group of rank rY ,• the canonical homomorphism

CH∗(XE)⊗Z CH∗(YE) → CH∗(XE × YE)

is an isomorphism.

Then∣∣∣∣ CH∗(XE × YE)

iE/F (CH∗(X × Y ))

∣∣∣∣ ≤ ∣∣∣∣ CH∗(XE)

iE/F (CH∗(X))

∣∣∣∣rY ·∣∣∣∣ CH∗(YE(X))


∣∣∣∣rX .

Proof. Let A = CH∗(XE), AR = iE/F (CH∗(X)) and

I =⊕p>0

CHp(XE) = F1CH∗(XE) .

Let B = CH∗(YE). By our assumption, we have CH∗(XE⊗YE) ≃ A⊗ZB. Wedenote by R the image of the composition

CH∗(X × Y ) → CH∗(XE ⊗ YE) ≃ A⊗Z B .

Clearly, all conditions of Lemma 3.3 hold. Moreover,∣∣∣∣ CH∗(XE × YE)

iE/F (CH∗(X × Y ))

∣∣∣∣ = ∣∣∣∣A⊗Z B

R

∣∣∣∣ and

∣∣∣∣ CH∗(XE)

iE/F (CH∗(X))

∣∣∣∣ = ∣∣∣∣ AAR∣∣∣∣ .

By Corollary 4.3 we have∣∣∣∣ A⊗Z B

R + (I ⊗Z B)

∣∣∣∣ = ∣∣∣∣ CH∗(YE(X))


∣∣∣∣ .To complete the prove it suffices to apply Lemma 3.3.

5. The group TorsCH2(Xψ ×XD)

Lemma 5.1 (see [32, (2.1)]). Let ψ be a (2n + 1)-dimensional quadratic

form over a separably closed field. Set Xdef= Xψ and d

def= dimX = 2n − 1.

Then for all 0 ≤ p ≤ d the group CHp(X) is canonically isomorphic to Z (forother p the group CHp(X) is trivial). Moreover,

• if 0 ≤ p < n, then CHp(X) = Z · hp, where h ∈ CH1(X) denotes theclass of a hyperplane section of X;


• if n ≤ p ≤ d, then CHp(X) = Z · ld−p, where ld−p denotes the class of alinear subspace in X of dimension d− p, besides 2ld−p = hp.

Corollary 5.2. Let ψ be a (2n+ 1)-dimensional quadratic form over Fand let X = Xψ. Then

• CH∗(X) is a free abelian group of rank 2n,• if 0 ≤ p < n then |CHp(X)/CH

p(X)| = 1,

• if n ≤ p ≤ 2n− 1 then |CHp(X)/CHp(X)| ≤ 2,

• |CH∗(X)/CH∗(X)| ≤ 2n.

Lemma 5.3. Let D be a central simple F -algebra of exponent 2 and ofdegree 8. Let E/L/F be field extensions such that indDL = 4 and indDE = 1.Let Y = SB(D). For any 0 ≤ p ≤ dimY = 7, the group CHp(YE) is canonicallyisomorphic to Z. Moreover, the image of the homomorphism iE/L : CHp(YL) →CHp(YE) ≃ Z contains 1 if p = 0, 4; 2 if p = 1, 2, 5, 6; 4 if p = 3, 7.

Proof. Since degD = 8 and indDE = 1, YE is isomorphic to P7E. Hence,

the group CHp(YE) ∼= CHp(P7E) (where p = 0, . . . , 7) is generated by the class

hp of a linear subspace ([14]).The rest part of Lemma is contained in [34, Theorem]. For convenience of

the reader, we also give a direct construction of the elements required. Let usdenote by CH

p(YL) the image of the composition

CHp(YL)iE/L−−−→ CHp(YE) ≃ Z .

The class of YL itself gives 1 ∈ CH0(YL). Let ξ be the tautological line bundle

on the projective space P7E ≃ YE. Since expD = 2, the bundle ξ⊗2 is defined

over F and, in particular, over L. Its first Chern class gives 2 ∈ CH1(YL).

Since indDL = 4, the bundle ξ⊕4 is defined over L. Its second Chern class

gives 6 ∈ CH2(YL).

1 Thus 2 ∈ CH2(YL). The third Chern class of ξ⊕4 gives

4 ∈ CH3(YL). The fourth Chern class of ξ⊕4 gives 1 ∈ CH

4(YL). Finally,

taking the product of the cycles constructed in codimensions 1, 2, and 3 withthe cycle of codimension 4, one gets the cycles of codimensions 5, 6, and 7required.

Corollary 5.4. Under the condition of Lemma 5.3, we have

|CH∗(YE)/iE/L(CH∗(YL))| ≤ 256 .

Proof.7∏p=0

|CHp(YE)/iE/L(CHp(YL))| ≤ 1 · 2 · 2 · 4 · 1 · 2 · 2 · 4 = 256 .

Proposition 5.5. Let D be a central division F -algebra of degree 8 andexponent 2. Let ψ be a 5-dimensional quadratic F -form. Suppose that DF (ψ)

is not a skewfield. Then TorsG∗K(Xψ ×XD) = 0.

1In fact, it is enough only to know that the Grothendieck classes of the bundles ξ⊗2 andξ⊕4 are in K(YL) what can be also seen from the computation of the K-theory.

6. THE GROUP H3(F (ψ,D)/F ) 131

Proof. Let X = Xψ and Y = XD. Corollary 5.2 shows that CH∗(XF ) isa free abelian group of rank rX = 4 and |CH∗(XF )/iF /F (CH

∗(X))| ≤ 22 = 4.Since D is a division algebra of degree 8 and DF (ψ) is not division algebra,

it follows that indDF (X) = 4. Applying Corollary 5.4 to the case L = F (X),E = F (X), we have |CH∗(YF (X))/iF (X)/F (X)(CH

∗(YF (X)))| ≤ 256.Since YF = SB(DF ) ≃ P7

F, the group CH∗(YF ) is a free abelian of rank

rY = 8 and CH∗(XF ) ⊗ CH∗(YF ) ≃ CH∗(XF × YF ) (see [12, Proposition14.6.5]). Thus all conditions of Proposition 4.4 hold for X, Y , E = F and wehave ∣∣∣∣ CH∗(XF × YF )

iF /F (CH∗(X × Y ))

∣∣∣∣ ≤ 48 · 2564 = 248.

Using [69, Theorem 4.1 of §8] and [82, Theorem 9.1], we get a natural(with respect to extensions of F ) isomorphism

K(X × Y ) ≃ K((F×3 × C)⊗F (F×4 ×D×4)

)≃

≃ K(F×12 × C×4 ×D×12 × (C ⊗F D)×4

)where C

def= C0(ψ) is the even Clifford algebra of ψ. Note that C is a central

simple F -algebra of the degree 22. Since DF (ψ) is not a skewfield, [55, Theorem1] states that D ≃ C⊗F B with some central division F -algebra B. Therefore,indC = degC = 22 and indC ⊗D = indB = degB = 2. Hence∣∣∣∣ K(XF × YF )

iF /F (K(X × Y ))

∣∣∣∣ = (indC)4 · (indD)12 · (indC ⊗D)4 = 22·4+3·12+1·4 = 248 .

Applying Proposition 2.4 to the variety X × Y and E = F , we have

|TorsG∗K(X × Y )| =|CH∗(XF × YF )/iF /F (CH

∗(X × Y ))||K(XF × YF )/iF /F (K(X × Y ))|

≤ 248

248= 1 .

Therefore, TorsG∗K(X × Y ) = 0.

Applying Lemma 1.5.1 we get the following

Corollary 5.6. Under the condition of Proposition 5.5, the groupCH2(Xψ ×XD) is torsion-free.

6. The group H3(F (ψ,D)/F )

Proposition 6.1 ([5, Satz 5.6]). Let ψ be a quadratic F -form of dimen-sion ≥ 5. The group H3(F (ψ)/F ) is non-trivial iff ψ is a neighbor of ananisotropic 3-Pfister form.

Proposition 6.2 (see [66, Proposition 4.1 and Remark 4.1]). Let D be acentral division F -algebra of exponent 2. Suppose that D is decomposable (inthe tensor product of two proper subalgebras). Then

H3(F (D)/F ) = [D] ∪H1(F ) .

Lemma 6.3. If D and D′ are Brauer equivalent central simple F -algebras,then the function fields F (D) and F (D′) are stably birational equivalent.


Proof. Since the algebras DF (D′) and D′F (D) are split, the field extensions

F (D,D′)/F (D′) and F (D,D′)/F (D)

are pure transcendental. Therefore each of the field extensions F (D)/F andF (D′)/F is stably birational equivalent to the extension F (D,D′)/F .

Corollary 6.4. Fix a quadratic F -form ψ and integers i, j ∈ Z. For anycentral simple F -algebra D, the groups H i(F (D)/F ), H i(F (D)/F,Q/Z(j)),H i(F (ψ,D)/F ), H i(F (ψ,D)/F,Q/Z(j)) depend only on the Brauer class ofD.

Lemma 6.5. Let D be a central simple F -algebra of exponent 2 and let ψbe a quadratic F -form. The group H3(F (ψ,D)/F,Q/Z(2)) is annihilated by2.

Proof. Let ψ0 be a 3-dimensional subform of ψ. Clearly,

H3(F (ψ,D)/F,Q/Z(2)) ⊂ H3(F (ψ0, D)/F,Q/Z(2)) .Therefore, it suffices to show that the latter cohomology group is annihilatedby 2. Replacing ψ0 by the quaternion algebra C0(ψ0), we come to a statementcovered by [24, Lemma A.8].

Corollary 6.6. In the conditions of Lemma 6.5, one has

H3(F (ψ,D)/F,Q/Z(2)) = H3(F (ψ,D)/F ) .

Proposition 6.7. Let D be an exponent 2 central simple F -algebra andlet ψ be a quadratic F -form of dimension ≥ 3. Suppose that indDF (ψ) < indD.Then ψ is not a 3-Pfister neighbor and there is an isomorphism

H3(F (ψ,D)/F )

H3(F (ψ)/F ) + [D] ∪H1(F )≃ TorsCH2(Xψ ×XD) .

Proof. By Proposition 2.2 of Chapter 4, there is an isomorphism

H3(F (ψ,D)/F,Q/Z(2))H3(F (ψ)/F,Q/Z(2)) +H3(F (D)/F,Q/Z(2))

≃

≃ TorsCH2(Xψ ×XD)

pr∗ψ TorsCH2(Xψ) + pr∗D TorsCH2(XD)

.

By Corollary 6.6, we have H3(F (ψ,D)/F,Q/Z(2)) = H3(F (ψ,D)/F ); byLemma 2.8 of Chapter 4, we have H3(F (ψ)/F,Q/Z(2)) = H3(F (ψ)/F ); andby [24, Lemma A.8], we have H3(F (D)/F,Q/Z(2)) = H3(F (D)/F ).

Let D′ be a division algebra Brauer equivalent to D. By Corollary 6.4, wehave H3(F (D)/F ) = H3(F (D′)/F ); by Proposition 1.1 of Chapter 2, we haveTorsCH2(XD) ≃ TorsCH2(XD′). Since D′

F (ψ) is no more a skewfield, there

is a homomorphism of F -algebras C0(ψ) → D′ ([84, Theoreme 1], see also[56, Theorem 2]). Although the algebra C0(ψ) is not always central simple, it

7. COROLLARIES 133

always contains a non-trivial subalgebra central simple over F . Therefore, D′

is decomposable, what implies H3(F (D′)/F ) = [D]∪H1(F ) (Proposition 6.2)and TorsCH2(XD′) = 0 (Proposition 5.3 of Chapter 1). Finally, the existenceof a homomorphism C0(ψ) → D′ implies that ψ is not a 3-Pfister neighbor;therefore TorsCH2(Xψ) = 0 ([32, Theorem 6.1]).

Corollary 6.8. Let D be a central division F -algebra of degree 8 andexponent 2. Let ψ be a 5-dimensional quadratic F -form. Suppose that DF (ψ)

is not a skewfield. Then H3(F (ψ,D)/F ) = [D] ∪H1(F ).

Proof. It is a direct consequence of Proposition 6.7, Corollary 5.6, andProposition 6.1.

Theorem 6.9. Let D be a central division F -algebra of degree 8 and ex-ponent 2. Let ψ be a quadratic form of dimension ≥ 5. Suppose that DF (ψ) isnot a skewfield. Then H3(F (ψ,D)/F ) = [D] ∪H1(F ).

Proof. Let ψ0 be a 5-dimensional subform of ψ. Applying Corollary 6.8,we have [D] ∪ H1(F ) ⊂ H3(F (ψ,D)/F ) ⊂ H3(F (ψ0, D)/F ) = [D] ∪ H1(F ).Hence H3(F (ψ,D)/F ) = [D] ∪H1(F ).

Corollary 6.10. Let ϕ ∈ I2(F ) be a 8-dimensional quadratic form suchthat indC(ϕ) = 8. Let D be a degree 8 central simple algebra such that c(ϕ) =[D]. Let ψ be a quadratic form of dimension ≥ 5 such that ϕF (ψ) is isotropic.Then

1) D is a division algebra;2) DF (ψ) is not a division algebra;3) H3(F (ψ,D)/F ) = [D] ∪H1(F ).

7. Corollaries

In this section we demonstrate two applications of Corollary 6.10. Namely,we give very short proofs of the following two theorems:

Theorem 7.1 ([47]). Let ϕ ∈ I2(F ) be an 8-dimensional quadratic formsuch that indC(ϕ) = 8. Let ψ be a quadratic form of dimension ≥ 5 such thatϕF (ψ) is isotropic. Then there exists a half-neighbor ϕ∗ of ϕ such that ψ ⊂ ϕ∗.

Theorem 7.2 ([10, Corollary 9.3]). Let ϕ ∈ I2(F ) be a quadratic formsuch that indC(ϕ) ≥ 8. Let A be an algebra such that c(ϕ) = [A]. ThenϕF (A) /∈ I4(F (A)). In particular, ϕF (A) is not hyperbolic. Moreover, if dimϕ =8 then ϕF (A) is anisotropic.

We need several lemmas.

Lemma 7.3. Let ϕ ∈ I2(F ) be a 8-dimensional quadratic form and let Dbe an algebra such that c(ϕ) = [D]. Then ϕF (D) ∈ GP3(F (D)).

Proof. We have c(ϕF (D)) = c(ϕ)F (D) = [DF (D)] = 0. Hence ϕF (D) ∈I3(F (D)). Since dimϕ = 8, we are done.


Lemma 7.4. Let ϕ, ϕ∗ ∈ I2(F ) be 8-dimensional quadratic forms such thatc(ϕ) = c(ϕ∗) = [D], where D is a 3-quaternion division algebra. Suppose thatthere is a quadratic form ψ of dimension ≥ 5 such that the forms ϕF (ψ,D) andϕ∗F (ψ,D) are isotropic. Then ϕ and ϕ∗ are half-neighbors.

Proof. Lemma 7.3 implies that ϕF (ψ,D), ϕ∗F (ψ,D) ∈ GP3(F (ψ,D)). By the

assumption of the lemma, ϕF (ψ,D) and ϕ∗F (ψ,D) are isotropic. Hence ϕF (ψ,D) and

ϕ∗F (ψ,D) are hyperbolic. Thus ϕ, ϕ∗ ∈ W (F (ψ,D)/F ).

Let τ = ϕ ⊥ ϕ∗. Clearly τ ∈ W (F (ψ,D)/F ). Since c(τ) = c(ϕ) + c(ϕ∗) =[D] + [D] = 0, we have τ ∈ I3(F ). Thus e3(τ) ∈ H3(F (ψ,D)/F ). It followsfrom Corollary 6.10 that e3(τ) ∈ [D]∪H1(F ). Hence there exists s ∈ F ∗ suchthat e3(τ) = [D] ∪ (s). We have e3(τ) = [D] ∪ (s) = c(ϕ) ∪ (s) = e3(ϕ ⟨⟨s⟩⟩).Since ker(e3 : I3(F ) → H3(F )) = I4(F ), we have τ ≡ ϕ ⟨⟨s⟩⟩ (mod I4(F )).Therefore ϕ+ ϕ∗ = τ ≡ ϕ ⟨⟨s⟩⟩ = ϕ− sϕ (mod I4(F )). Hence ϕ∗ + sϕ ∈ I4(F ).Hence ϕ and ϕ∗ are half-neighbors.

The following statement was pointed out by Laghribi ([47]) as an easyconsequence of the index reduction formula [55].

Lemma 7.5. Let ψ be a quadratic form of dimension ≥ 5 and D be adivision 3-quaternion algebra. Suppose that DF (ψ) is not division algebra. Thenthere exists an 8-dimensional quadratic form ϕ∗ ∈ I2(F ) such that ψ ⊂ ϕ∗ andc(ϕ∗) = [D].

Proof of Theorem 7.1. LetD be 3-quaternion algebra such that c(ϕ) =[D]. Since indC(ϕ) = 8, it follows that D is a division algebra. Since ϕF (ψ)

is isotropic, DF (ψ) is not a division algebra. It follows from Lemma 7.5 thatthere exists an 8-dimensional quadratic form ϕ∗ ∈ I2(F ) such that ψ ⊂ ϕ∗ andc(ϕ∗) = [D]. Obviously, all conditions of Lemma 7.4 hold. Hence ϕ and ϕ∗ arehalf-neighbors.

Lemma 7.6. Let D be a division 3-quaternion algebra over F . Then thereexist a field extension E/F and an 8-dimensional quadratic form ϕ∗ ∈ I2(E)with the following properties:

(i) DE is a division algebra,(ii) c(ϕ∗) = [DE],(iii) ϕ∗

E(D) is anisotropic.

Proof. Let ϕ ∈ I2(F ) be an arbitrary F -form such that c(ϕ) = [D].Let K = F (X,Y, Z) and γ = ϕK ⊥ ⟨⟨X, Y, Z⟩⟩ be a K-form. Let K =K0, K1, . . . , Kh; γ0, γ1, . . . , γh be a generic splitting tower of γ.

Since γ ≡ ϕK (mod I3(K)), we have c(γ) = c(ϕK) = [DK ]. Since K/F ispurely transcendental, indDK = indD = 8. Hence indC(γ) = 8. It followsfrom Corollary 1.2.3 that there exists s such that dim γs = 8 and indC(γs) = 8.We set E = Es, ϕ

∗ = γs.We claim that the condition (i)–(iii) of the lemma hold. Since c(ϕ∗) =

c(γE) = c(ϕE) = [DE], condition (ii) holds. Since [DE] = c(ϕ∗) = c(γs), wehave indDE = indC(γs) = 8 and thus condition (i) holds.

8. A GENERALIZATION 135

Now we only need to verify that (iii) holds. Let M0/F be an arbitraryfield extension such that ϕM0 is hyperbolic. Let M = M0(X,Y, Z). We haveγM = ϕM ⊥ ⟨⟨X, Y, Z⟩⟩M . Clearly ⟨⟨X,Y, Z⟩⟩ is anisotropic over M . SinceϕM is hyperbolic, we have (γM)an = ⟨⟨X, Y, Z⟩⟩M and hence dim(γM)an =8. Therefore dim(γM)an = dim γs. By Lemma 1.2.1, we see that the fieldextension ME/M = MKs/M is purely transcendental. Hence dim(γME)an =dim(γM)an = 8. Since (ϕ∗

ME)an = (γME)an, we see that ϕ∗ME is anisotropic.

Since ϕM is hyperbolic, it follows that [DM ] = c(ϕM) = 0. Hence [DME] = 0and therefore the field extensionME(D)/ME is purely transcendental. Henceϕ∗ME(D) is anisotropic. Therefore ϕ

∗E(D) is anisotropic.

Lemma 7.7. Let ϕ, ϕ∗ ∈ I2(F ) be 8-dimensional quadratic forms such thatc(ϕ) = c(ϕ∗) = [D], where D is a 3-quaternion division algebra. Suppose thatϕ∗F (D) is anisotropic. Then ϕF (D) is anisotropic.

Proof. Suppose at the moment that ϕF (D) is isotropic. Then letting ψdef=

ϕ∗, we see that all conditions of Lemma 7.4 hold. Hence ϕ and ϕ∗ are half-neighbors, i.e., there exists s ∈ F ∗ such that ϕ∗ + sϕ ∈ I4(F ). Thereforeϕ∗F (D) + sϕF (D) ∈ I4(F (D)). Since ϕF (D) is hyperbolic, we see that ϕ∗

F (D) ∈I4(F (D)). By the Arason-Pfister Hauptsatz, we see that ϕ∗

F (D) is hyperbolic.So we get a contradiction to the assumption of the lemma.

Proposition 7.8. Let ϕ ∈ I2(F ) be an 8-dimensional quadratic form suchthat indC(ϕ) = 8. Let A be an algebra such that c(ϕ) = [A]. Then ϕF (A) isanisotropic.

Proof. Let D be a 3-quaternion algebra such that c(ϕ) = [D]. SinceindC(ϕ) = 8, D is a division algebra. Let E/F and ϕ∗ be such that in Lemma7.6. All conditions of Lemma 7.7 hold for E, ϕE, ϕ

∗, and DE. Therefore ϕE(D)

is anisotropic. Hence ϕF (D) is anisotropic. Since [A] = c(ϕ) = [D], the fieldextension F (A)/F is stably isomorphic to F (D)/F (Lemma 6.3). ThereforeϕF (A) is anisotropic.

Proof of Theorem 7.2. Suppose at the moment that ϕF (A) ∈ I4(F (A)).Since indC(ϕ) ≥ 8, it follows that dimϕ ≥ 8. By Corollary 1.2.3 there ex-ists a field extension E/F such that dim(ϕE)an = 8, indC(ϕE) = 8. Sincedim(ϕE)an = 8 and ϕE(A) ∈ I4(E(A)), the Arason-Pfister Hauptsatz showsthat ((ϕE)an)E(A) is hyperbolic. We get a contradiction to Proposition 7.8.

8. A generalization

In this section, we generalize Corollary 5.6 and Theorem 6.9 to the case ofarbitrary indD.

Theorem 8.1. Let D be a central simple F -algebra of exponent 2. Let ψbe a quadratic form of dimension ≥ 5. Suppose that indDF (ψ) < indD. Then

TorsCH2(Xψ ×XD) = 0 and H3(F (ψ,D)/F ) = [D] ∪H1(F ).


Proof. By Proposition 6.7, there is a surjection

H3(F (ψ,D)/F )

[D] ∪H1(F )� TorsCH2(Xψ ×XD) .

Thus, it suffices to prove the second formula of Theorem.Proving the second formula, we may assume that dimψ = 5 (compare to

the proof of Theorem 6.9) and D is a division algebra (Corollary 6.4). Underthese assumptions, we can write down D as the tensor product C0(ψ) ⊗F B(using [55, Theorem 1]). In particular, we see that C0(ψ) is a division algebra,i.e. indC0(ψ) = degC0(ψ) = 4.

If degD < 8, then D ≃ C0(ψ). In this case, ψF (D) is a 5-dimensional qua-dratic form with trivial Clifford algebra; therefore ψF (D) is isotropic; by thisreason, the field extension F (ψ,D)/F (D) is pure transcendental and conse-quently H3(F (ψ,D)/F (D)) = 0. It follows that

H3(F (ψ,D)/F ) = H3(F (D)/F ) = [D] ∪H1(F ) ,

where the last equality holds by Proposition 6.2.If degD > 8, then indB ≥ 4. Applying the index reduction formula [78,

Theorem 1.3], we get

indC0(ψ)F (D) = min{indC0(ψ), indB} = 4 .

Therefore ψF (D) is not a 3-Pfister neighbor and by Proposition 6.1 the groupH3(F (ψ,D)/F (D)) is trivial. Thus once again

H3(F (ψ,D)/F ) = H3(F (D)/F ) = [D] ∪H1(F ) .

Finally, if degD = 8, then we are done by Theorem 6.9 and Proposition6.7.

Remark 8.2. A computation of the group H3(F (ψ,D)/F ) in some othercases not covered here is given in Chapters 3 and 7.

CHAPTER 7

Isotropy of 8-dimensional quadratic forms over functionfields of quadrics

Let F be a field of characteristic different from 2 and ϕ be an anisotropic8-dimensional quadratic form over F with trivial determinant. We study thelast open cases in the problem of describing the quadratic forms ψ such thatϕ becomes isotropic over the function field F (ψ).


0. Introduction

Let F be a field of characteristic different from 2 and let ϕ and ψ be twoanisotropic quadratic forms over F . An important problem in the algebraictheory of quadratic forms is to find conditions on ϕ and ψ so that ϕF (ψ) isisotropic. More precisely, one studies the question whether the isotropy of ϕover F (ψ) is standard in a sense.

In this chapter we consider the case where ϕ is an 8-dimensional anisotropicquadratic form with trivial determinant. Necessity of certain sufficient con-ditions for isotropy of ϕ over F (ψ) was studied by A. Laghribi; we call theisotropy, caused by one of these conditions, L-standard:

Definition. Let ϕ and ψ be anisotropic quadratic forms such that ϕF (ψ)

is isotropic. Besides we suppose that dimϕ = 8 and detϕ = 1. We say thatthe isotropy of ϕF (ψ) is L-standard, if at least one of the following conditionsholds:

• there exists a half-neighbor ϕ∗ of ϕ such that ψ ⊂ ϕ∗;• there exists a 5-dimensional subform ϕ0 ⊂ ϕ with the following twoproperties:


Otherwise, we say that the isotropy is non-L-standard.

In the case when dimψ ≥ 5, the isotropy of ϕF (ψ) is always L-standard([46], [47], see also Chapter 6). The main result of this chapter is the following

Theorem. Let ϕ be an anisotropic 8-dimensional quadratic form withdetϕ = 1 and ψ be a 4-dimensional quadratic form with detψ = 1. Sup-pose that ϕF (ψ) is isotropic. Then the isotropy of ϕF (ψ) is L-standard exceptthe case indC(ϕ) = ind(C(ϕ)⊗F C0(ψ)) = 4.

137


For the exceptional case of the theorem see Corollary 7.4.For the case where dimψ = 3 or where dimψ = 4 and detψ = 1 see §8.

1. Terminology and notation

1.1. Quadratic forms. Mainly, we use notation of [48] and [76]. How-ever there is certain slight difference: we denote by ⟨⟨a1, . . . , an⟩⟩ the n-foldPfister form ⟨1,−a1⟩⊗ · · ·⊗⟨1,−an⟩. We denote by Pn(F ) the set of all n-foldPfister forms; GPn(F ) is the set of the forms similar to a form from Pn(F ). Forany field extension L/F , we denote by Pn(L/F ) the set of forms from Pn(F )hyperbolic over L; GPn(L/F ) is the set of the forms similar to a form fromPn(L/F ).

We recall that a quadratic form ψ is called a (Pfister) neighbor (of a Pfisterform π), if it is similar to a subform in π and dimϕ > 1

2dimπ. Two quadratic

forms ϕ and ϕ∗ are half-neighbors, if dimϕ = dimϕ∗ and there exists s ∈ F ∗

such that the sum ϕ⊥sϕ∗ is similar to a Pfister form.For a quadratic form ϕ of dimension ≥ 3, we denote by Xϕ the projective

variety given by the equation ϕ = 0. We set F (ϕ) = F (Xϕ) if dimϕ ≥ 3;

F (ϕ) = F (√d) if dimϕ = 2 and d = d±ϕ = 1; and F (ϕ) = F otherwise.

1.2. Algebras. We consider only finite-dimensional F -algebras.For a central simple F -algebra D, we denote by deg(D), [D], and exp(D)

respectively the degree of D, the class of D in the Brauer group Br(F ), andthe exponent of D, i.e., the order of [D] in the Brauer group.

For a simple F -algebra A, we denote by ind(A) the Schur index of A. Foran algebra B of the form B = A×· · ·×A with simple A, we set indB = indA.

Let ϕ be a quadratic form. We denote by C(ϕ) the Clifford algebra of ϕ.By C0(ϕ) we denote the even part of C(ϕ). Note that for any quadratic F -formψ and any central simple F -algebra D, the index of C0(ψ)⊗FD is well-defined.

If ϕ ∈ I2(F ) then C(ϕ) is a central simple algebra. Its class [C(ϕ)] in theBrauer group Br2(F ) is denoted by c(ϕ).

Let D be a central simple algebra. We denote by XD the Severi-Brauervariety of D and by F (D) the function field F (XD). For another centralsimple F -algebra D′ and for a quadratic F -form ψ of dimension ≥ 3, we set

F (D′, D)def= F (XD′ ×XD) and F (ψ,D)

def= F (Xψ ×XD).

1.3. Cohomology groups. By H∗(F ) we denote the graded ring of Ga-

lois cohomology H∗(F,Z/2Z) def= H∗(Gal(Fsep/F ),Z/2Z).

For n = 0, 1, 2, 3, there is a homomorphism en : In(F ) → Hn(F ) which isuniquely determined by the condition en(⟨⟨a1, . . . , an⟩⟩) = (a1, . . . , an) (see [5]).The homomorphism en is surjective and ker en = In+1(F ) for n = 0, 1, 2, 3 (see[53], [62], and [71]).

We also work with the cohomology groups Hn(F,Q/Z(i)), (i = 0, 1, 2),defined by B. Kahn (see [29]). For n = 1, 2, 3, the group Hn(F ) is naturallyidentified with Tors2H

n(F,Q/Z(n− 1)).

2. THE GROUPS H3(F (ψ,D)/F ) AND I3(F (ψ,D)/F ) 139

1.4. K-theory and Chow groups. For a smooth algebraic F -varietyX, its Grothendieck ring is denoted by K(X). This ring is supplied with thefiltration by codimension of support (also called the topological filtration).

We fix an algebraic closure F of the base field F and denote by X theF -variety XF . If the variety X is projective homogeneous, we identify K(X)with a subring of K(X) via the restriction homomorphism which is injectiveby [65].

For a ring (or a group) with filtration A, we denote by G∗A the adjointgraded ring (resp., the adjoint graded group).

There is a canonical surjective homomorphism of the graded Chow ringCH∗(X) onto G∗K(X), its kernel consists only of torsion elements and is trivialin the 0-th, 1-st, and 2-nd graded components ([81, §9]).

Let X1 and X2 be two smooth F -varieties. For any x1 ∈ K(X1) andx2 ∈ K(X2), we denote by x1�x2 the product pr

∗1(x1) · pr∗2(x2) ∈ K(X1×X2)

where pr∗1 and pr∗2 are the pull-backs with respect to the projections pr1 andpr2 of X1×X2 on X1 and X2 respectively. For an OX1-module F1 and an OX2-module F2, we denote by F1 � F2 the tensor product pr∗1(F1)⊗OX pr

∗2(F2).

1.5. Relative groups. Let Φ be an arbitrary functor on the category offields (of characteristic = 2) with values in the category of abelian groups. Fora field extension L/F we use the notation Φ(L/F ) for ker(Φ(F ) → Φ(L)).Here is a list of examples that we need in this chapter: W (L/F ), In(L/F ),Hn(L/F ), and Hn(L/F,Q/Z(i)).

2. The groups H3(F (ψ,D)/F ) and I3(F (ψ,D)/F )

Proposition 2.1. Let D be a central simple F -algebra of exponent 2 andψ be a quadratic form of dimension ≥ 3. Then there exists a natural isomor-phism

H3(F (ψ,D)/F )

H3(F (ψ)/F ) +H3(F (D)/F )≃ TorsCH2(Xψ ×XD)


where pr∗ψ and pr∗D are the pull-backs with respect to the projection prψ andprD of Xψ ×XD to Xψ and XD.

Proof. By Proposition 2.2 of Chapter 4, the factor group

H3(F (ψ,D)/F,Q/Z(2))H3(F (ψ)/F,Q/Z(2)) +H3(F (D)/F ),Q/Z(2))

is isomorphic to

TorsCH2(Xψ ×XD)


.


Now it is sufficient to apply isomorphisms

H3(F (ψ)/F,Q/Z(2)) = H3(F (ψ)/F ),

H3(F (D)/F,Q/Z(2)) = H3(F (D)/F ),

H3(F (ψ,D)/F,Q/Z(2)) = H3(F (ψ,D)/F ).

(see e.g. Lemma 2.8 of Chapter 4, [24, Lemma A.8], Corollary 6.6 of Chapter6).

Corollary 2.2. Let D be a biquaternion algebra and ψ be a 4-dimensionalquadratic form. Then there exists a natural isomorphism

H3(F (ψ,D)/F )

H3(F (ψ)/F ) +H3(F (D)/F )≃ TorsCH2(Xψ ×XD).

In particular, 2 · TorsCH2(Xψ ×XD) = 0.

Proof. Since dimψ = 4, we have TorsCH2(Xψ) = 0 (see e.g. [32, Theo-rem 6.1] or Lemma 2.4 of Chapter 4). By Proposition 5.3 of Chapter 1, we haveTorsCH2(XD) = 0. To complete the proof it is sufficient to apply Proposition2.1.

Lemma 2.3. Let ψ be a quadratic form of dimension ≥ 3. Then

1. the map e3 : P3(F (ψ)/F ) → H3(F (ψ)/F ) is surjective;2. I3(F (ψ)/F ) + I4(F ) = P3(F (ψ)/F ) + I4(F ).

Proof. 1. Really, we have to verify that the set H3(F (ψ)/F ) consists ofsymbols. This fact is proved in [5, Satz 5.6].2. Follows from Item 1 and from injectivity of e3 : I3(F )/I4(F ) → H3(F ).

Lemma 2.4. Let D be a biquaternion algebra and q be an Albert form ofD. Then

1. H3(F (D)/F ) = [D] ∪H1(F );2. I3(F (D)/F ) + I4(F ) = {q ⟨⟨s⟩⟩ | s ∈ F ∗}+ I4(F );3. the map e3 : I3(F (D)/F ) → H3(F (D)/F ) is surjective.

Proof. 1. See [66, Corollary 4.5].2. Obviously e3(q ⟨⟨s⟩⟩) = [D] ∪ (s). Now, it is sufficient to apply Item 1 andinjectivity of e3 : I3(F )/I4(F ) → H3(F ).3. Follows from Items 1 and 2.

Lemma 2.5. Let D be a biquaternion algebra and ψ be a quadratic formof dimension ≥ 3. Then the natural homomorphism

I3(F (ψ,D)/F ) + I4(F )

I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F )→ H3(F (ψ,D)/F )

H3(F (ψ)/F ) +H3(F (D)/F )

is injective.In particular, the condition H3(F (ψ,D)/F ) = H3(F (ψ)/F )+H3(F (D)/F )

implies that I3(F (ψ,D)/F ) ⊂ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F ).

3. THE GROUP K(Xψ ×XD) 141

Proof. Obvious consequence of the following facts:a) I3(F )/I4(F ) → H3(F ) is injective;b) I3(F (ψ)/F )) → H3(F (ψ)/F ) is surjective (Lemma 2.3);c) I3(F (D)/F )) → H3(F (D)/F ) is surjective (Lemma 2.4).

Corollary 2.6. Let D be a biquaternion algebra and ψ be a quadraticform of dimension ≥ 3 such that TorsCH2(Xψ ×XD) = 0. Then

I3(F (ψ,D)/F ) ⊂ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F ) .

Proof. Follows from Lemma 2.5 and Proposition 2.1.

3. The group K(Xψ ×XD)

In this section, ψ is a quadratic F -form of dimension 4, D is a biquaternionF -algebra.

Lemma 3.1. Consider the tensor product K(Xψ)⊗ZK(XD) together withthe filtration induced by the topological filtrations on K(Xψ) and K(XD). Theadjoint graded group G∗(K(Xψ)⊗K(XD)) is torsion-free.

Proof. The adjoint graded group G∗K(Xψ) is torsion-free (see e.g [32]).The adjoint graded group G∗K(XD) is torsion-free as well (see [34, Example]).We have a surjection

G∗K(Xψ)⊗G∗K(XD) � G∗(K(Xψ)⊗K(XD)) .

The left-hand side term is a finitely generated torsion-free abelian group, i.e.a free abelian group of finite rank. This rank coincides with the rank of theright-hand side term. Therefore, the map is an isomorphism.

We consider the subgroup K(Xψ)�K(XD) of K(Xψ ×XD) together withthe filtration induced by the topological filtration on K(Xψ ×XD).

Lemma 3.2. The homomorphism K(Xψ)⊗K(XD) → K(Xψ)�K(XD) isan isomorphism of groups with filtrations.

Proof. It is a homomorphism of groups respecting the filtrations. First ofall let us check that it is an isomorphism of groups, regardless the filtrations. Itis evidently an epimorphism. So, we only have to check the injectivity. Sincefor any extension of the base field F , the restriction homomorphism on theproductK(Xψ)⊗K(XD) is injective, it suffices to check the injectivity in a splitsituation. However, if D splits, then XD is isomorphic to a projective space;therefore the map K(Xψ)⊗K(XD) → K(Xψ)�K(XD) is an isomorphism.

To finish the proof, it is suffices to show that the homomorphism of theadjoint graded groups is injective. Consider the commutative diagram

G∗(K(Xψ)⊗K(XD)) −−−→ G∗(K(Xψ)�K(XD))x xG∗(K(Xψ)⊗K(XD)) −−−→ G∗(K(Xψ)�K(XD))


The left-hand side arrow is injective since the group G∗(K(Xψ) ⊗K(XD)) istorsion-free (Lemma 3.1) The upper arrow is injective because XD is isomor-phic to a projective space. Therefore the bottom arrow is injective too.

Corollary 3.3. The group G∗(K(Xψ)�K(XD)) is torsion-free.

We denote by C the even Clifford algebra C0(ψ). Let U(2) be Swan’s sheafU on the quadric Xψ ([82, §6]), twisted twice. It has a structure of C-module.Let J be the canonical sheaf on the Severi-Brauer variety XD ([69, §8.4]). Ithas a structure of D-module.

Put F def= U(2)�J . It is a sheaf on Xψ ×XD with a structure of C ⊗F D-

module. Denote by f the homomorphism K(C⊗D) → K(Xψ×XD) given bythe functor of tensor multiplication by F over C ⊗D.

Put G def= U(2)�J ⊗3. It is a sheaf onXψ×XD with a structure of C⊗FD

⊗3-module. Consider the homomorphism K(C ⊗D⊗3) → K(Xψ ×XD) given bythe functor of tensor multiplication by G over C ⊗ D⊗3. Since the algebraD⊗2 is split, the group K(C ⊗ D⊗3) is canonically isomorphic (via Morita

equivalence) to K(C ⊗ D). Denote by g the the composition K(C ⊗ D)∼→

K(C ⊗D⊗3) → K(Xψ ×XD).

Lemma 3.4. 1. The homomorphism

K(C ⊗D)⊕2 → K(Xψ ×XD)/(K(Xψ)�K(XD)) ,

induced by the homomorphisms f + g, is surjective.2. If C ⊗D is a skewfield, then K(Xψ ×XD) = K(Xψ)�K(XD).

Proof. 1. Using Swan’s computation of the K-theory for quadrics [82,Theorem 9.1] (with U(2) instead of U) and a generalized Peyre’s version[66, Proposition 3.1] of Quillen’s computation of K-theory for Severi-Brauerschemes [69, Theorem 4.1 of §8], we get an isomorphism

K(F )⊕4 ⊕K(C)⊕2 ⊕K(D)⊕4 ⊕K(C ⊗D)⊕2 ≃ K(Xψ ×XD)

such that the image of K(F )⊕4 ⊕K(C)⊕2 ⊕K(D)⊕4 is contained in K(Xψ)�K(XD) and the summand K(C⊗D)⊕2 is mapped into K(Xψ×XD) via f +g.Therefore, K(C⊗D)⊕2 → K(Xψ×XD)/(K(Xψ)�K(XD)) is an epimorphism.2. If the algebra C ⊗ D is a skewfield, then its class generates the groupK(C⊗D). The images of this class under f , g are F ,G ∈ K(Xψ)�K(XD).

Corollary 3.5. If C ⊗D is a skewfield, then the group G∗K(Xψ ×XD)is torsion-free.

Proof. By Lemma 3.4, K(Xψ ×XD) = K(Xψ) �K(XD). By Corollary3.3, the group G∗(K(Xψ)�K(XD)) is torsion-free.

4. The group TorsCH2(Xψ ×XD)

Theorem 4.1. Let D be a biquaternion algebra and ψ be an anisotropic4-dimensional quadratic form with detψ = 1. Then the group CH2(Xψ ×XD)is torsion-free except (possibly) the following two cases:

4. THE GROUP TorsCH2(Xψ ×XD) 143

(1) indC0(ψ)⊗D = indD = 4,(2) indC0(ψ)⊗D = 2.

Proof. Set Cdef= C0(ψ) and s

def= ind(C⊗D). The possible values of s are

1, 2, 4, 8.Assume that s = 8. Since detψ = 1, it follows that C ⊗D is a skewfield.

Therefore, the group CH2(Xψ × XD) ≃ G2K(Xψ × XD) is torsion-free byCorollary 3.5.

Now assume s = 1. Then ind(C0(ψF (D))) = 1. Therefore the quadraticform ψF (D) is isotropic. Hence the extension F (ψ,D)/F (D) is purely tran-scendental and H3(F (ψ,D)/F ) = H3(F (D)/F ). Now, Corollary 2.2 showsthat TorsCH2(Xψ ×XD) = 0.

Finally, assume s = 4 and indD = 4. Then the biquaternion algebra Dis Brauer equivalent to a quaternion F -algebra D′. Since TorsCH2(Xψ ×XD)depends only on the Brauer class of D (see e.g. Chapter 6 ), we may replaceD by D′. Let ψ′ be a 3-dimensional quadratic F -form such that C0(ψ

′) ≃ D′.The Severi-Brauer variety XD′ is isomorphic to the conic Xψ′ . Since the tensorproduct C0(ψ) ⊗ C0(ψ

′) has index 4, it is a division algebra. Therefore, byCorollary 4.4 of Chapter 4, the group CH2(Xψ ×Xψ′) is torsion-free.

Remark 4.2. The assumption ψ is anisotropic is not essential: if ψ isisotropic, then

H3(F (ψ,D)/F ) = H3(F (D)/F )

and therefore the group CH2(Xψ × XD) is torsion-free as well (by Corollary2.2).

Proposition 4.3. Let D be a biquaternion algebra and let ψ be a 4-dimensional quadratic form with detψ = 1. Then the group TorsCH2(Xψ ×XD) is equal to zero or isomorphic to Z/2.

Proof. Since we already know that the torsion in the group CH2(Xψ×XD)is annihilated by 2 (Corollary 2.2), it suffices to show that the torsion is cyclic.

Once again we set Cdef= C0(ψ). According to Theorem 4.1, it suffices to

consider the case where ind(C ⊗D) equals 2 or 4.Consider the quotient K(Xψ×XD)/(K(Xψ)�K(XD)) with the filtration

induced by the topological filtration on K(Xψ × XD). Since in the exactsequence of the adjoint graded groups

0 → G∗(K(Xψ)�K(XD)) → G∗K(Xψ ×XD) →→ G∗(K(Xψ ×XD)/(K(Xψ)�K(XD))) → 0

the left-hand side term is torsion-free (Corollary 3.3), we have an injection

TorsG∗K(Xψ ×XD) ↪→ G∗(K(Xψ ×XD)/(K(Xψ)�K(XD))) .

Since CH2(Xψ × XD) ≃ G2K(Xψ × XD), it suffices to show that the groupG2(K(Xψ ×XD)/(K(Xψ)�K(XD))) is cyclic.


Denote by h ∈ K(Xψ) the class of a general hyperplane section of Xψ. Letξ ∈ K(XD) be the class of the tautological linear bundle on the projectivespace XD. Note that for any i ≥ 0, the multiple (indD⊗i) · ξi of ξi belongs toK(XD). Thus ξ

i ∈ K(XD) for i even and 4ξi ∈ K(XD) for i odd.The algebra C ⊗D is simple. Therefore, its Grothendieck group is cyclic.

By Lemma 3.4, it follows that the quotient K(Xψ ×XD)/(K(Xψ)�K(XD))

is generated by two elements, namely by (s/2)x and (s/2)y, where xdef= (4 +

2h + h2) � ξ, ydef= (4 + 2h + h2) � ξ3, and s

def= indC ⊗ D (we use here the

equality [U(2)] = 4 + 2h+ h2 ∈ K(Xψ), [32, Lemma 3.6]).We have a congruence x ≡ h2 � (ξ − 1) − h � (ξ − 1)2 (mod K(Xψ) �

K(XD)). The right-hand side element belongs to K(Xψ×XD)(2), because it is

inK(Xψ×XD)(2) andK(Xψ×XD)

(2) = K(Xψ×XD)(2)∩K(Xψ×XD) (see [74,

Lemme 6.3, (i)]). Therefore, (s/2)x ∈ (K(Xψ × XD)/(K(Xψ) � K(XD)))(2)

(take into account that the coefficient s/2 is an integer).We also have another congruence modulo K(Xψ)�K(XD):

y − x ≡(h2 � (ξ − 1)− h� (ξ − 1)2

)·(1� (ξ2 − 1)

).

The right-hand side element is in K(Xψ × XD)3 as a product of an element

in K(Xψ ×XD)(2) and the element 1� (ξ2 − 1) ∈ K(Xψ ×XD)

(1). Therefore,(s/2)(y − x) ∈ (K(Xψ × XD)/(K(Xψ) � K(XD)))

(3) and it follows that thegroup G2(K(Xψ × XD)/(K(Xψ) � K(XD))) is generated by (s/2)x. So, inparticular, this group is cyclic.

Remark 4.4. The assumption detψ = 1 is not essential: if detψ = 1,then H3(F (ψ,D)/F ) = H3(F (ψ′, D)/F ) and H3(F (ψ)/F ) = H3(F (ψ′)/F ),where ψ′ is an arbitrary 3-dimensional subform of ψ (Lemma 5.2 of Chapter4). Therefore,

CH2(Xψ ×XD) ≃ CH2(Xψ′ ×XD) ≃ CH2(XD′ ×XD)

where D′ is the even Clifford algebra of ψ′. The group TorsCH2(XD′ × XD)is zero or Z/2 according to Theorem 6.1 of Chapter 2.

Theorem 4.5. Suppose that ψ = ⟨−x,−y, xy, d⟩ (with d /∈ F ∗2) and D =(x, y)⊗ (u, v) where x, y, d, u, v ∈ F ∗. Then TorsCH2(Xψ ×XD) = 0.

Proof. The even Clifford algebra of the quadratic form ψ is isomorphic tothe quaternion algebra (x, y)F (

√d). Therefore, the tensor product C0(ψ) ⊗ D

is Brauer equivalent to the quaternion algebra (u, v)F (√d) and in particular

has index 2 or 1. In the case, where the index is 1, we are done by Theorem4.1. Let us assume the index equals to 2. It suffices to show that the element

xdef= h2�(ξ−1)−h�(ξ−1)2 is in K(Xψ×XD)

(3) (see the proof of Proposition4.3).

By definition, the element h ∈ K(Xψ) is the class of a hyperplane sectionof the quadric Xψ. This hyperplane section is the quadric Xψ′ determined

by a 3-dimensional subform ψ′ of ψ. Clearly, x is equal to the image of x′def=

5. STANDARD ISOTROPY IN THE CASE TorsCH2(Xψ ×XD) = 0 145

h′�(ξ−1)−1�(ξ−1)2 ∈ K(Xψ′×XD) under the push-forward homomorphismK(Xψ′ ×XD) → K(Xψ ×XD). Moreover, x′ ∈ K(Xψ′ ×XD)

(2), because x′ ∈K(Xψ′×XD)

(2) and K(Xψ′×K(XD))(2) = K(Xψ′×XD)

(2)∩K(Xψ′×K(XD)).Since the codimension of Xψ′ × XD in Xψ × XD equals to 1, it follows thatx ∈ K(Xψ ×XD)

(3).

Remark 4.6. The assumption d ∈ F ∗2 is not essential: if d ∈ F ∗2, then

TorsCH2(Xψ × XD) ≃ TorsCH2(XC0(ψ′) × XD), where ψ′ def= ⟨−x,−y, xy⟩;

since C0(ψ′) ≃ (x, y), we have indC0(ψ

′)⊗D ≤ 2; therefore, by Theorem 6.1of Chapter 2, the latter group is zero.

5. Standard isotropy in the case TorsCH2(Xψ ×XD) = 0

Definition 5.1. We say that (ϕ,D, q) is a special triple if the followingconditions hold:

1) ϕ is an 8-dimensional anisotropic form with detϕ = 1,2) D is a biquaternion algebra,3) q is an Albert form,4) [D] = c(ϕ) = c(q) ∈ Br2(F ).

In this section we need the following

Theorem 5.2. Let ϕ be an anisotropic 8-dimensional quadratic form withdetϕ = 1 and let D be an algebra such that c(ϕ) = [D]. Then ϕF (D) isanisotropic.

Proof. See [46, Theoreme 4] and [10, Corollary 9.3], see also Theorem7.2 of Chapter 6.

Our study of isotropy of 8-dimensional forms over function field of quadricsis based on the following assertion.

Proposition 5.3. Let (ϕ,D, q) be a special triple and ψ be a quadraticform. Then

1. The following two conditions are equivalent:(i) ϕ+ q ∈ I3(F (ψ,D)/F );(ii) ϕF (ψ) is isotropic.

2. The following two conditions are equivalent:(i) ϕ+ q ∈ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F );(ii) there exists a 5-dimensional Pfister neighbor ϕ0 such that ϕ0 ⊂ ϕ


Proof. (1i)⇒(1ii). Condition (1i) implies that the quadratic form (ϕ ⊥q)F (ψ,D) is hyperbolic. Since qF (D) is hyperbolic, it follows that ϕF (ψ,D) is

hyperbolic. Let Edef= F (ψ). We see that ϕE(D) is hyperbolic. Theorem 5.2

implies that ϕE is isotropic, i.e., condition (1ii) holds.(1ii)⇒(1i). Suppose that ϕF (ψ) is isotropic. Since c(ϕ) = c(q), it follows thatϕ + q ∈ I3(F ). Hence it is sufficient to prove that ϕF (ψ,D) and qF (ψ,D) are hy-perbolic. The form qF (ψ,D) is hyperbolic because qF (D) is. Since c(ϕ) = [D], we


have ϕF (ψ,D) ∈ I3(F (ψ,D)). Since ϕF (ψ) is isotropic, we have dim(ϕF (ψ,D))an <8. The Arason-Pfister Hauptsatz shows that ϕF (ψ,D) is hyperbolic.(2i)⇒(2ii). By Lemmas 2.3 and 2.4, there exist π ∈ P3(F (ψ)/F ) and s ∈ F ∗

such thatϕ+ q ≡ π + q ⟨⟨s⟩⟩ (mod I4(F )).

We have ϕ+sq ≡ π (mod I4(F )). Since π ∈ P3(F ), πF (π) is hyperbolic. Hence(ϕ + sq)F (π) ≡ πF (π) ≡ 0 (mod I4(F )). Since dim(ϕ + sq) ≤ 8 + 4 < 16, theArason-Pfister Hauptsatz shows that (ϕ + sq)F (π) is hyperbolic. Hence thereexists a form γ such that (ϕ ⊥ sq)an = πγ. Clearly 0 < 8 − 6 ≤ dim(ϕ ⊥sq)an ≤ 8 + 6 < 16. This implies that dim γ = 1, i.e., there exists k ∈ F ∗ suchthat γ = ⟨k⟩. Thus ϕ + sq = kπ. Therefore (ϕ ⊥ −kπ) = −sq. Hence ϕ andkπ contain a common subform of dimension

dimϕ+ dimπ − dim q

2=

8 + 8− 6

2= 5.

Let us denote such a form by ϕ0. Since dimϕ0 = 5 and ϕ0 ⊂ kπ, it follows thatϕ0 is a Pfister neighbor of π. Since π ∈ P3(F (ψ)/F ), it follows that (ϕ0)F (ψ)

is isotropic.(2ii)⇒(2i). Let ϕ0 be a 5-dimensional quadratic form such as in (2ii). Bythe assumption, there exists π ∈ GP3(F ) such that ϕ0 ⊂ π. Since (ϕ0)F (ψ) isisotropic, it follows that π ∈ GP3(F (ψ)/F ).

Since ϕ0 ⊂ ϕ and ϕ0 ⊂ π, there exist 3-dimensional quadratic forms ρ′, ρ′′

such that ϕ = ϕ0 ⊥ ρ′ and π = ϕ0 ⊥ ρ′′. We set ρ = ρ′′ ⊥ −ρ′. Clearlydim ρ = 6. In the Witt ring W (F ) we have ρ = ρ′′ − ρ′ = π− ϕ. In particular,ρ ∈ I2(F ). Hence ρ is an Albert form.

We have c(ρ) = c(π)+ c(ϕ) = 0+ c(ϕ) = c(q). Hence ρ is similar to q ([28,Theorem 3.12]). Let s ∈ F ∗ be such that ρ = sq. We have π − ϕ = ρ = sq.Hence ϕ = π − sq. Therefore

ϕ+ q = π + q ⟨⟨s⟩⟩ ∈ GP3(F (ψ)/F ) + [q] · I(F ) ⊂⊂ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F ).

Corollary 5.4. Let (ϕ,D, q) be a special triple and ψ be a quadratic formof dimension ≥ 3. Suppose that TorsCH2(Xψ ×XD) = 0. Then the followingconditions are equivalent:

(1) ϕF (ψ) is isotropic,(2) there exits a 5-dimensional Pfister neighbor ϕ0 such that ϕ0 ⊂ ϕ and

(ϕ0)F (ψ) is isotropic.

Proof. Obvious in view of Proposition 5.3 and Corollary 2.6.

6. The group H3(F (ψ,D)/F ) in the case ind(C0(ψ)⊗F D) = 2

In this section we study the group H3(F (ψ,D)/F ) in the case where ψ is a4-dimensional quadratic form with a non-trivial discriminant, D is a biquater-nion division F -algebra, and ind(C0(ψ)⊗F D) = 2.

6. THE GROUP H3(F (ψ,D)/F ) IN THE CASE ind(C0(ψ)⊗F D) = 2 147

Let ddef= detψ. By our assumption, d /∈ F ∗2. Replacing ψ by a similar

form, we can suppose ψ = ⟨−a,−b, ab, d⟩ with a, b ∈ F ∗. Let Ldef= F (

√d).

By our assumption, we have ind((a, b) ⊗F D)L = 2. Hence there exists aquaternion F -algebra Q such that (a, b)L + [DL] = [QL] in Br2(L) (see [42,Proposition 16.2]). Let us write Q in the form Q = (r, s) with r, s ∈ F ∗. Weset ψ′ = ⟨−r,−s, rs, d⟩.

Let q be an Albert form corresponding to D.

Lemma 6.1. There exist k, k′ ∈ F ∗ such that kψ + k′ψ′ + q ∈ I3(F ).

Proof. Since (a, b)L+[DL] = [QL] = (r, s)L, it follows that (a, b)+(r, s)+[D] ∈ Br2(L/F ). Hence there exist k ∈ F ∗ such that (a, b)+(r, s)+[D] = (d, k).

Let k′def= −1 and ϕ

def= kψ+ k′ψ′ + q. We claim that ϕ ∈ I3(F ). To prove this,

it is sufficient to verify that ϕ ∈ I2(F ) and c(ϕ) = 0. We have

ϕ = kψ + k′ψ′ + q = k ⟨−a,−b, ab, d⟩ − ⟨−r,−s, rs, d⟩+ q =

= k(⟨⟨a, b⟩⟩ − ⟨⟨d⟩⟩)− (⟨⟨r, s⟩⟩ − ⟨⟨d⟩⟩) + q =

= k ⟨⟨a, b⟩⟩+ ⟨⟨d, k⟩⟩ − ⟨⟨r, s⟩⟩+ q.

Hence ϕ ∈ I2(F ) and c(ϕ) = (a, b) + (d, k) + (r, s) + c(q) = (a, b) + (d, k) +(r, s) + [D] = 0.

Definition 6.2. LetD be a biquaternion algebra and ψ be a 4-dimensionalquadratic form such that detψ = 1 and ind(C0(ψ)⊗F D) = 2. We denote byΓ(ψ,D) the set defined as follows

{γ ∈ I3(F ) | there exist k, k′, l ∈ F ∗ such that γ = kψ + k′ψ′ + lq},

where q is an Albert form corresponding to D and ψ′ is a 4-dimensional qua-dratic form satisfying the following two properties: detψ′ = detψ and C0(ψ

′)is Brauer-equivalent to C0(ψ)⊗F D.

Remark 6.3. 1. The set Γ(ψ,D) does not depend on the choice of qand ψ′: indeed, the condition on q and ψ′ determines them uniquely upto similarity.

2. Lemma 6.1 shows the set Γ(ψ,D) is not empty.

Lemma 6.4. Γ(ψ,D) ⊂ I3(F (ψ,D)/F ).

Proof. Let γ = kψ + k′ψ′ + lq ∈ Γ(ψ,D). By the definition of Γ(ψ,D),we have γ ∈ I3(F ). Thus it is sufficient to prove that γF (ψ,D) is hyperbolic.We have dim(ψF (ψ))an ≤ 2 and dim(qF (D))an = 0. Therefore dim(γF (ψ,D))an =dim((kψ ⊥ k′ψ′ ⊥ lq)F (ψ,D))an ≤ 2 + 4 + 0 = 6 < 8. Since γ ∈ I3(F ), theArason-Pfister Hauptsatz shows that γF (ψ,D) is hyperbolic.

Lemma 6.5. Let γ ∈ Γ(ψ,D), π ∈ P3(F (ψ)/F ), and s ∈ F ∗. Then thereexists γ′ ∈ Γ(ψ,D) such that γ′ ≡ γ + π + q ⟨⟨s⟩⟩ (mod I4(F )).


Proof. Let us write γ in the form γ = kψ + k′ψ′ + lq. Since π ∈P3(F (ψ)/F ), there exists r ∈ F ∗ such that π = ψ ⟨⟨r⟩⟩.

We have kψ + π ≡ kψ − kπ = kψ − kψ ⟨⟨r⟩⟩ = rkψ (mod I4(F )) andlq + q ⟨⟨s⟩⟩ ≡ lq − lq ⟨⟨s⟩⟩ = lsq (mod I4(F )). Therefore

γ + π + q ⟨⟨s⟩⟩ = kψ + k′ψ′ + lq + π + q ⟨⟨s⟩⟩ ≡ rkψ + k′ψ′ + lsq (mod I4(F )).

Now it is sufficient to set γ′ = rkψ + k′ψ′ + lsq.

Corollary 6.6.

Γ(ψ,D) + I4(F ) = Γ(ψ,D) + I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F ) .

Proof. Obvious in view of Lemmas 6.5, 2.3, and 2.4.

Lemma 6.7. The following conditions are equivalent:

(1) I3(F (ψ,D)/F ) ⊂ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F );(2) Γ(ψ,D) ⊂ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F );(3) there exists γ ∈ Γ(ψ,D) such that

γ ∈ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F );

(4) Γ(ψ,D) contains a hyperbolic form, i.e. 0 ∈ Γ(ψ,D);(5) there exist x, y, u, v, d ∈ F ∗ such that ψ ∼ ⟨−x,−y, xy, d⟩ and D ∼=

(x, y)⊗F (u, v);(6) TorsCH2(Xψ ×XD) = 0;(7) H3(F (ψ,D)/F ) = H3(F (ψ)/F ) +H3(F (D)/F ).

Proof. (1)⇒(2). Obvious in view of Lemma 6.4.(2)⇒(3). Obvious, because Γ(ψ,D) is not empty.(3)⇒(4). Condition (3) implies that 0 ∈ Γ(ψ,D)+I3(F (ψ)/F )+I3(F (D)/F )+I4(F ). It follows from Corollary 6.6 that 0 ∈ Γ(ψ,D) + I4(F ). Hence thereexists γ = kψ + k′ψ′ + lq ∈ Γ(ψ,D) such that γ ∈ I4(F ). Since dim γ =4 + 4 + 6 = 14 < 16, the Arason-Pfister Hauptsatz shows that γ = 0.(4)⇒(5). Let γ = kψ+k′ψ′+ lq be a hyperbolic form. We have (kψ ⊥ lq)an =−k′ψ′

an. Therefore kψ and −lq contain a common subform of dimension

1

2(dimψ + dim q − dimψ′) =

1

2(4 + 6− 4) = 3.

Let us denote such a 3-dimensional form by τ . Let x, y ∈ F ∗ be such thatτ ∼ ⟨−x,−y, xy⟩. Thus ⟨−x,−y, xy⟩ is similar to a subform of ψ and similarto a subform of q. Let d = detψ. Since ⟨−x,−y, xy⟩ is similar to a subform ofψ it follows that ⟨−x,−y, xy, d⟩ is similar to ψ. Since ⟨−x,−y, xy⟩ is similarto a subform of the Albert form q, it follows that there exist u, v ∈ F ∗ suchthat q is similar to ⟨−x,−y, xy, u, v,−uv⟩. Then [D] = c(q) = (x, y) + (u, v).Therefore D ∼= (x, y)⊗F (u, v).(5)⇒(6). See Theorem 4.5.(6)⇒(7). See Proposition 2.1.(7)⇒(1). See Lemma 2.5.

7. MAIN THEOREM 149

Proposition 6.8. Let D be a biquaternion algebra and let ψ be a 4-dimen-sional quadratic form such that detψ = 1 and ind(D⊗F C0(ψ)) = 2. Then forany γ ∈ Γ(ψ,D) one has

H3(F (ψ,D)/F ) = H3(F (ψ)/F ) +H3(F (D)/F ) + e3(γ)H0(F ).

Proof. By Lemma 6.4, the element e3(γ) belongs to H3(F (ψ,D)/F ). Ifthe group CH2(Xψ ×XD) is torsion-free, then, by Proposition 2.1, we have

H3(F (ψ,D)/F ) = H3(F (ψ)/F ) +H3(F (D)/F )

and the proof is complete. If TorsCH2(Xψ × XD) = 0, Lemma 6.7 showsthat γ /∈ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F ). It follows from Lemma 2.5 thate3(γ) /∈ H3(F (ψ)/F ) +H3(F (D)/F ). To complete the proof it is sufficient toapply Proposition 4.3 and Proposition 2.1.

Corollary 6.9.

I3(F (ψ,D)/F ) ⊂ I3(F (ψ)/F ) + I3(F (D)/F ) + Γ′(ψ,D) + I4(F ),

where Γ′(ψ,D) = Γ(ψ,D) ∪ {0}.

Proof. Let τ ∈ I3(F (ψ,D)/F ). Choose an element γ ∈ Γ(ψ,D). ByProposition 6.8, either the element e3(τ) or the element e3(τ − γ) is in

H3(F (ψ)/F ) +H3(F (D)/F ) .

It remains to apply Lemma 2.5.

Proposition 6.10. Let λ ∈ I3(F (ψ,D)/F ). Then at least one of thefollowing conditions holds

1) λ ∈ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F );2) λ ∈ Γ(ψ,D) + I4(F ).

Proof. Obvious in view of Corollaries 6.9 and 6.6.

7. Main theorem

Theorem 7.1. Let ϕ be an anisotropic 8-dimensional quadratic form withdetϕ = 1 and let ψ be a 4-dimensional quadratic form with detψ = 1. Supposethat ϕF (ψ) is isotropic and let the case indC(ϕ) = ind(C(ϕ)⊗F C0(ψ)) = 4 beexcepted. Then the isotropy of ϕF (ψ) is L-standard.

Proof. In the case where indC(ϕ) = 8, the theorem is proved in Chapter3. Thus we can suppose that indC(ϕ) ≤ 4. Then there exists a biquaternionalgebra D such that c(ϕ) = [D]. Let q be an Albert form corresponding to D.Clearly (ϕ,D, q) is a special triple .

By Corollary 5.4, we can suppose that TorsCH2(Xψ ×XD) = 0. Theorem4.1 asserts that at least one of the following conditions holds:

1) ind(D ⊗ C0(ψ)) = indD = 4,2) ind(D ⊗ C0(ψ)) = 2.


If ind(D⊗C0(ψ)) = 2, then all the conditions of Definition 6.2 hold. Thus wehave a well-defined set Γ(ψ,D).

By Proposition 5.3, we have ϕ+ q ∈ I3(F (ψ,D)/F ). By Proposition 6.10,we see that at least one of the following condition holds:

1) ϕ+ q ∈ I3(F (ψ)/F ) + I3(F (D)/F ) + I4(F );2) ϕ+ q ∈ Γ(ψ,D) + I4(F ).

In the first case, Proposition 5.3 shows that there exists a 5-dimensionalPfister neighbor ϕ0 such that ϕ0 ⊂ ϕ and (ϕ0)F (ψ) is isotropic. This impliesthat the isotropy of ϕF (ψ) is L-standard.

Therefore, we may assume that ϕ+ q ∈ Γ(ψ,D) + I4(F ). Thus there existk, k′, l ∈ F ∗ (and a 4-dimensional quadratic form ψ′) such that

ϕ+ q ≡ kψ + k′ψ′ + lq (mod I4(F )).

Since kψ + k′ψ′ + lq ∈ I3(F ) it follows that

kψ + k′ψ′ + lq ≡ l(kψ + k′ψ′ + lq) (mod I4(F )).

Therefore

ϕ+ q ≡ lkψ + lk′ψ′ + q (mod I4(F )).

Thus ϕ ≡ lkψ + lk′ψ′ (mod I4(F )). Let ϕ∗ def= ψ ⊥ kk′ψ′. Since ϕ ≡ lkϕ∗

(mod I4(F )) and dimϕ∗ = 8, it follows that ϕ and ϕ∗ are half-neighbors.Since ψ ⊂ ϕ∗, the isotropy is L-standard and the proof is complete.

Corollary 7.2. Let ϕ be an anisotropic 8-dimensional quadratic formwith detϕ = 1 and let ψ be a 4-dimensional quadratic form with detψ = 1.Suppose that ϕF (ψ) is isotropic but the isotropy is not L-standard. Then ϕ canbe written in the form ϕ = π1 ⊥ π2 with π1, π2 ∈ GP2(F ).

Proof. Since detϕ = 1, it is sufficient to verify that ϕ contains a 4-dimensional quadratic form with trivial determinant. Suppose at the mo-ment that ϕ contains no 4-dimensional quadratic form with trivial determi-nant. Then [25, Theorem 6.1] implies that there exists a homomorphism ofF -algebras, C0(ψ) → C0(ϕ). Since detϕ = 1 and dimϕ = 8, there exists a 3-quaternion algebra A such that C0(ϕ) has the form A×A and C(ϕ) ≃M2(A).Thus we get a homomorphism C0(ψ) → A which is injective because C0(ψ)is a simple algebra. Then ind(C0(ψ) ⊗F A) = 2. Since M2(A) ≃ C(ϕ), wehave ind(C0(ψ) ⊗F C(ϕ)) = 2. Theorem 7.1 implies that isotropy of ϕF (ψ) isL-standard. The contradiction obtained completes the proof.

It is a natural question if there exists an example of non-L-standard isotropy.One way to find non-L-standard isotropy is based on the following

Lemma 7.3. Let q be a 6-dimensional quadratic form and let ψ be a 4-dimensional quadratic form over a field k. Suppose that q is a k(ψ)-minimal

form (see definition in [20]). Let Fdef= k((t)), ϕ

def= q ⊥ t ⟨1, det(q)⟩. Then the

form ϕF (ψ) is isotropic, but the isotropy is not L-standard.

8. ISOTROPY OVER THE FUNCTION FIELD OF A CONIC 151

Proof. The proof of this lemma is absolutely analogous to the proof ofTheorem 5.1 of Chapter 3 and we omit it.

Corollary 7.4. There exist a field F , an 8-dimensional quadratic formϕ ∈ I2(F ), and a 4-dimensional F -form ψ with nontrivial determinant suchthat ϕF (ψ) is isotropic, but the isotropy is not L-standard.

Proof. By Corollary 15.4 of Chapter 5, there exists a field k, a quadraticform q of dimension 6, and a quadratic form ψ of dimension 4 over a fieldk such that q is a k(ψ)-minimal form. Now the required result follows fromLemma 7.3.

8. Isotropy over the function field of a conic

In this section we still assume that ϕ is an anisotropic 8-dimensional qua-dratic form of trivial determinant. We are interested in the question when ϕis isotropic over the function field of a quadratic form ψ.

For the forms ψ of dimension ≥ 5, the question was studied in [47] and [46].The case dimψ = 4, d±ψ = 1 is done in the previous section. Thus it sufficesto consider only two cases: dimψ = 3 or ψ ∈ GP2(F ). Since the function fieldof a form ψ ∈ GP2(F ) is stably birational equivalent to the function field ofan arbitrary 3-dimensional subform of ψ (see e.g. Lemma 5.2 of Chapter 4),it suffices to handle only one of these cases.

Let us consider the case where ψ ∈ GP2(F ).

Theorem 8.1. Let ϕ be an anisotropic 8-dimensional quadratic form (wedo not assume detϕ = 1) and let ψ ∈ GP2(F ). Then the form ϕF (ψ) is isotropicif and only if at least one of the following conditions holds:

a) there exists a 10-dimensional form ϕ∗ such that ψ ⊂ ϕ∗ and sϕ ≡ ϕ∗

(mod I4(F )) for suitable s ∈ F ∗;b) there exists a 5-dimensional subform ϕ0 ⊂ ϕ with the following two

properties:– the form ϕ0 is a Pfister neighbor,– the form (ϕ0)F (ψ) is isotropic.

Proof. First, suppose that ϕF (ψ) is isotropic. The excellence propertyof the field extension F (ψ)/F implies that there exist a 6-dimensional formτ and a form γ such that ϕ − τ = γψ (see [20, Proof of Proposition 1.1]).Comparing dimensions, one can see that 1 ≤ dim γ ≤ 3. First, consider the

case where dim γ is odd (i.e., dim γ = 1 or 3). We set sdef= d±γ. Clearly, γ ≡ ⟨s⟩

(mod I2(F )). Hence γψ ≡ sψ (mod I4(F )). Since ϕ−τ = γψ, it follows that

ϕ = τ+γψ ≡ τ+s·ψ ≡ s(sτ+ψ) (mod I4(F )). Setting ϕ∗ def= sτ ⊥ ψ, one can

see that condition a) holds. Now, suppose that dim γ is odd, i.e., dim γ = 2.Then γψ ∈ GP3(F ). We have ϕ − γψ = τ . Therefore, the quadratic forms ϕand γψ contain a common subform of dimension 1

2(dimϕ+dim γψ−dim τ) =

12(8 + 8 − 6) = 5. Let us denote that 5-dimensional form by ϕ0. Clearly,

condition b) holds.


Now, suppose that at least one of conditions a) and b) holds. We haveto verify that ϕF (ψ) is isotropic. It is obvious in the case where condition b)holds. Suppose now that condition a) holds. Since ψ ⊂ ϕ∗ and dimϕ∗ = 10,one has dim(ϕ∗

F (ψ))an ≤ 6. Therefore dim((ϕ ⊥ −sϕ∗)F (ψ))an ≤ 8 + 6 =

14 < 16. Since ϕ ⊥ −sϕ∗ ∈ I4(F ), the Arason-Pfister Hauptsatz implies thatdim((ϕ ⊥ −sϕ∗)F (ψ))an is hyperbolic. Hence (ϕF (ψ))an = (sϕ∗

F (ψ))an. Therefore

dim(ϕF (ψ))an ≤ 6. Hence ϕF (ψ) is isotropic.

Corollary 8.2. Let ϕ be an anisotropic 8-dimensional quadratic formwith detϕ = 1 and let ψ be a 4-dimensional quadratic form with detψ = 1 (orψ is a 3-dimensional form). Suppose that ϕF (ψ) is isotropic but the isotropy isnot L-standard. Then ind(C(ϕ)⊗ C0(ψ)) = 4.

Proof. We can assume that ψ ∈ GP2(F ) (if dimψ = 3, then we replaceψ by ψ ⊥ ⟨detψ⟩) . Let us suppose that ind(C(ϕ)⊗C0(ψ)) = 4. By Theorem8.1, there exist a 10-dimensional form ϕ∗ and an coefficient s ∈ F ∗ such thatψ ⊂ ϕ∗ and sϕ ≡ ϕ∗ (mod I4(F )). Then ϕ∗ can be written in the formϕ∗ = ψ ⊥ q. Clearly, q is an Albert form and c(q) = c(ϕ∗) + c(ψ). ThereforeindC(q) = ind(C(ϕ∗)⊗C(ψ)) = ind(C(ϕ)⊗C0(ψ)) = 4. Hence, q is isotropic.

Thus there exists a 4-dimensional form q such that qan = qan. Set ϕ∗ def= ψ ⊥ q.

Clearly, dim ϕ∗ = 8, ψ ⊂ ϕ∗ and sϕ ≡ ϕ∗ ≡ ϕ∗ (mod I4(F )). Therefore, theisotropy ϕF (ψ) is L-standard, a contradiction.

Remark 8.3. There are many examples of ϕ and ψ with ψ ∈ GP2(F )such that the isotropy of ϕF (ψ) is not L-standard. The condition of Theorem8.1 can be regarded as a modification of the notion of the L-standard isotropyfor the case ψ ∈ GP2(F ).

CHAPTER 8

A generalization of the Albert-Risman theorems

Let A be a central simple algebra of a prime degree p over a field F and letB1, . . . , Bp−1 be central simple F -algebras of degrees pn1 , . . . , pnp−1 . We showthat if every tensor product A⊗F Bi has zero divisors, then there exists a fieldextension E/F of degree ≤ pn1+···+np−1 which splits the algebras B1, . . . , Bp−1

as well as the algebra A. In the case p = 2, this statement was proved 1975by L. Risman ([70]); in the case p = 2 and n1 = 1, it is a classical theorem ofA. Albert (see [1] or [2]).

0. Introduction

A well-known theorem of A. Albert states (see [1] or [2]): if tensor productof two quaternion algebras has zero divisors, then the quaternion algebras canbe split by a common extension of the base field of degree ≤ 2.

1975 L. J. Risman gave the following generalization of Albert’s theorem([70, Theorem 1]): if tensor product of a degree 2n (where n ≥ 1) centralsimple algebra A and a quaternion algebra B has zero divisors, then A and Bpossess a common splitting field of degree ≤ 2n.

Attempts to find a generalization of Risman’s theorem to the case of anodd prime p was unsuccessful for a long time. Even worth: 1993 B. Jacob andA. R. Wadsworth ([27], see also Chapter 9) have shown that already Albert’stheorem has no generalization to the case of two degree p algebras. Two degreep central simple algebras A,B with no common splitting field of degree p, theyfound for every odd prime p, possessed the following property: for any integersi, j ≥ 0 the index of the tensor product A⊗i ⊗B⊗j was ≤ p.

We propose here the following generalization of Risman’s theorem:

Theorem 0.1. Let A be a central simple algebra of a prime degree p overa field F and let B1, . . . , Bp−1 be central simple F -algebras of degrees pn1, . . . ,

pnp−1. Set ndef= n1 + · · ·+ np−1 and suppose that for every i = 1, . . . , p− 1 the

tensor product A ⊗F Bi has zero divisors. Then there exists a field extensionE/F of degree ≤ pn which splits all the algebras A,B1, . . . , Bp−1.

In the particular case where n1 = · · · = np−1 = 1, Theorem 0.1 can beregarded as a generalization of Albert’s theorem. For instance, taking p = 3we get the following

Example 0.2. Let A,B,C be three degree 3 central simple algebras overa field F . Suppose that each of the two tensor products A ⊗ B and A ⊗ C

153

154 8. A GENERALIZATION OF THE ALBERT-RISMAN THEOREMS

has zero divisors. Then there exists a field extension E/F of degree ≤ 9 whichsplits all the three algebras A,B,C.

In fact, a general method of obtaining results of the type similar to Theorem0.1 is developed in Chapter 10. However this method allows to control degreesof common splitting fields of algebras only up to a prime to p factor. In orderto obtain the announced exact statement, we use here a refinement of thatmethod. It is based on the intersection theory and especially on the theoryof non-negative intersections developed in [12, Chapter 12] (see the proof ofProposition 1.2).

Terminology and notation. Let F be a field. We fix an algebraic closureF of F and, for any F -variety X, denote by X the F -variety XF . Let σ bea cycle on X. We denote by [σ] its class in the Chow group CH∗(X) and byσ the corresponding cycle on X. Sometimes, while working on X × Y , whereY is another F -variety, we denote, abusing notation, by σ as well the cycleσ × Y .

Degree deg(σ) of a simple 0-dimensional cycle σ is defined to be the degreeof its residue field over the base field. Degree of an arbitrary 0-dimensionalcycle σ =

∑ljσj, where lj are integers and σj are simple cycles, is defined as∑

lj deg(σj).A cycle σ =

∑ljσj (of any dimension) is called non-negative, if all the

integers lj are non-negative.

1. Preliminaries

In this section we prove two (independent) statements needed for the nextsection.

Lemma 1.1. For any integers n,m ≥ 0, let ϕ : Pn × Pm ↪→ Pnm+n+m

be the Segre imbedding. Denote by f ∈ CH1(Pn), g ∈ CH1(Pm), and h ∈CH1(Pnm+n+m) the classes of hyperplanes. Then

ϕ∗(h) = f + g ∈ CH1(Pn × Pm),

where ϕ∗ : CH∗(Pnm+n+m) → CH∗(Pn × Pm) is the pull-back homomorphism.

Proof. Denote by [x0 : · · · : xn], [y0 : · · · : ym], and [z0 : · · · : znm+n+m] thecoordinates in Pn, Pm, and Pnm+n+m. The Segre embedding ϕ is determinedby the rule

ϕ([x0 : · · · : xn]× [y0 : · · · : ym]) = [x0y0 : x0y1 : · · · : xnym−1 : xnym] .

The intersection of the hyperplane z0 = 0 with Pn × Pm has two transversalcomponents: one of them is determined in Pn × Pm by the equation x0 = 0and represents f while the other one is determined by the equation y0 = 0 andrepresents g.

2. THE PROOF 155

Proposition 1.2. Let T be a direct product of Severi-Brauer varietiesover a field F . Let σ and σ′ be non-negative cycles on T and let ψ : T ′ → T bea morphism of F -varieties. Then

1. there exists a non-negative cycle τ on T such that [τ ] = [σ] · [σ′] ∈CH∗(T );

2. there exists a non-negative cycle τ ′ on T ′ such that ψ∗([σ]) = [τ ′].

Proof. We define the cycles τ and τ ′ a la [12]: τ is the intersection productσ ∩ σ′ of σ and σ′; τ ′ is the pull-back ψ!(σ) of the cycle σ with respect to ψ.Note that the cycles τ and τ ′ are defined precisely, not only up to rationalequivalence.

Considering τ as a cycle on T and τ ′ as a cycle on T ′, we have [τ ] =[σ] · [σ′] ∈ CH∗(T ) and [τ ′] = ψ∗([σ]) ∈ CH∗(T ′). So, we only have to checkthat τ and τ ′ are non-negative. For this it suffices to check that the cycles τand τ ′ on the F -varieties T and T ′ are non-negative.

Since intersection products commute with flat pull-backs ([12, Theorem6.2]), we still have τ = σ∩ σ′ and τ ′ = ψ!(σ), where ψ is the morphism T ′ → Tobtained from ψ by the base change. Since the variety T is isomorphic to adirect product of projective spaces, the tangent bundle on T is generated bythe global sections (see [12, Examples 12.2.1a and 12.2.1c]). Therefore, by[12, Corollary 12.2], the cycles τ and τ ′ are non-negative.

2. The proof

In this section we prove Theorem 0.1.We denote by X, Y1, . . . , Yp−1, and T1, . . . , Tp−1 the Severi-Brauer varieties

of the algebras A, B1, . . . , Bp−1, and A⊗F B1, . . . , A⊗F Bp−1. We set

Ydef=

p−1∏i=1

Yi .

For every i, tensor product of ideals induces a closed imbedding ψi : X ×Yi ↪→ Ti which is a twisted form of the Segre imbedding Pp−1 × Ppni−1 ↪→Ppni+1−1.

Let f ∈ CH1(X), gi ∈ CH1(Yi), and hi ∈ CH1(Ti) be the classes of hyper-planes.

The algebra A⊗Bi (for every i) has degree pni+1 and zero divisors, so that

its index divides pni . Therefore, there exists a simple (and in particular non-

negative) pni-codimensional cycle σi on the variety Ti such that [σi] = hpni

i ∈CHpni (Ti) (see [6, §3]).

By Item 2 of Proposition 1.2, there exists a non-negative cycle τi on X×Yisuch that [τi] = ψ∗

i ([σi]). Since [σi] = hpni

i , it follows from Lemma 1.1 that[τi] = (f + gi)

pni ∈ CHpni (X × Yi).Applying Item 1 of Proposition 1.2 to the variety X × Y , we find a non-

negative cycle τ on X × Y such that [τ ] = [τ1] · · · [τp−1] ∈ CH∗(X × Y ). Notethat τ is a cycle of codimension pn1 + · · · + pnp−1 on a variety of dimension

156 8. A GENERALIZATION OF THE ALBERT-RISMAN THEOREMS

(p − 1) + (pn1 − 1) + · · · + (pnp−1 − 1) = pn1 + · · · + pnp−1 , so that it is a0-dimensional cycle. Moreover,

[τ ] = [τ1] · · · [τp−1] = (f + g1)pn1 · · · (f + gp−1)

pnp−1=

= pn · (fp−1gpn1−1

1 · · · gpnp−1−1p−1 ),

where the last equality holds because of the relation fp = 0. Since fp−1 is theclass of a rational point on X and since gp

ni−1i is the class of a rational point

on the variety Yi for every i, we get deg(τ) = pn.Let τ ′ be any simple cycle included in τ . Since the cycle τ is non-negative,

we have deg(τ ′) ≤ pn. The residue field E of τ ′ is a common splitting fieldof the algebras A,B1, . . . , Bp−1 and [E : F ] = deg(τ ′) ≤ pn. The proof ofTheorem 0.1 is complete.

CHAPTER 9

Linked algebras

Two central simple algebras A,B of a prime degree p over a field are calledlinked, if for any integers i, j ≥ 0 the index of A⊗i ⊗ B⊗j is at most p. Theyare called strongly linked, if they possess a common splitting field of a finitedegree (over the base field) not divisible by p2.

We show that for any two not strongly linked central simple algebras anextension of the base field can be made such that the algebras become linkedwhile still not being strongly linked.

0. Introduction

1931 A. Albert proved (see [1] or [2]): two quaternion division algebras canbe split by a common quadratic extension of the base field provided that theirtensor product has zero divisors.

Attempts to generalize the theorem of Albert to the case of an odd prime phave led 1987 to counter-examples of J.-P. Tignol and A. R. Wadsworth ([85,Proposition 5.1]), who constructed two degree p central division algebras A,Bwith zero divisors in A⊗ B and without common splitting field extensions ofdegree p.

Stronger counter-examples was obtained 1993 by B. Jacob and A. R. Wads-worth ([27]). Two degree p central division algebras A,B without commonsplitting field of degree p, they found, possessed the following property: forany integers i, j ≥ 0 the index of the tensor product A⊗i ⊗ B⊗j was ≤ p. Itwas in fact even shown that any common splitting field of the algebras A,Bhas degree divisible by p2 ([27, Remark 2]).

In this Chapter we show (see Theorem 1.1) that similar counter-examplescan be obtained by an appropriate base extension starting from any two degreep central simple algebras A,B provided that degree of every common splittingfield of A,B is divisible by p2 (what is guaranteed if e.g. the tensor productA ⊗ B is a division algebra). The proof is essentially different from that of[27].

Notation. For a smooth variety X over a field F , we denote by K(X) theGrothendieck group of X; by K(X)(n), where n ≥ 0, the n-codimensional termof the topological filtration on X (see [69, §7] for a definition of the topologicalfiltration); by CH∗(X) the Chow ring of X graded by codimension of cycles; byCH0(X) the 0-dimensional component of the Chow ring, i.e. the component

157

158 9. LINKED ALGEBRAS

CHdimX(X). For a central simple F -algebra A, we denote by SB(A) the Severi-Brauer variety of A and by SB(r, A) (for r ≥ 0) the generalized Severi-Brauervarieties (also called generic partial splitting varieties) of A.

1. The theorem

Throughout this Chapter, A,B are central simple algebras of a prime de-gree p over a field F . We call them linked, if ind(A⊗i ⊗F A

⊗j) ≤ p for anyi, j ≥ 0. The algebras A and B are called strongly linked, if there exists a finitefield extension E/F such that

• the algebras AE and BE are split and• the degree [E : F ] is not divisible by p2.

Clearly, strongly linked algebras are linked. By the theorem of Albert, theinverse is true for p = 2. For any odd p, we shall prove the following

Theorem 1.1. Let F be a field, p be an odd prime number, and A,B bedegree p central simple F -algebras. Suppose that A,B are not strongly linked.Then there exists a field extension F /F such that the algebras AF , BF arelinked but still not strongly linked.

One can take for F the function field F (T ) of the following product ofgeneralized Severi-Brauer varieties

Tdef= SB(p,A⊗B)× SB(p,A⊗2 ⊗B)× · · · × SB(p,A⊗p−1 ⊗B) .

2. Preliminary observations

We set Xdef= SB(A) × SB(B). Let L/F be an arbitrary common splitting

field extension of A,B.

Lemma 2.1. The algebras A,B are not strongly linked if and only if theimage of the restriction homomorphism CH0(X) → CH0(XL) is divisible byp2.

Proof. Note that since the algebras AL, BL are split, the variety XL isisomorphic to a product of two projective spaces. Therefore the degree mapdeg : CH0(XL) → Z is an isomorphism. Also note that the compositionCH0(X) → CH0(XL) → Z of the restriction and degree maps is the degreemap for CH0(X).

If the image of the restriction homomorphism CH0(X) → CH0(XL) is notdivisible by p2, then there exists a closed point on X of degree not divisibleby p2. The residue field of this point is a common splitting field for A and Bshowing that the algebras are strongly linked.

To prove the inverse implication, suppose the image of the restriction ho-momorphism CH0(X) → CH0(XL) is divisible by p2. Then the image of thedegree homomorphism deg : CH0(X) → Z is divisible by p2 as well. Let E/F beany finite field extension such that the algebras AE, BE are split. The varietyXE has then a rational point. Taking the transfer, we obtain a 0-dimensional

3. THE PROOF 159

cycle of degree [E : F ] on X. Consequently, [E : F ] is divisible by p2, i.e. thealgebras A,B are not strongly linked.

Lemma 2.2. If the algebras A,B are not strongly linked, then, for any notsimultaneously 0 integers 0 ≤ i, j < p, the index of tensor product A⊗i ⊗ B⊗j

is divisible by p.

Proof. Let i, j be any integers such that 0 ≤ i, j < p. Suppose that thetensor product A⊗i⊗B⊗j is split. If i = 0, then any splitting field of B splits Aas well; therefore the algebras A,B possess a common splitting field of degreep in this case, what contradicts to the assumption they are not strongly linked.Thus i = 0. The symmetric argument shows that j = 0 as well.

Lemma 2.3. Suppose that, for any not simultaneously 0 integers 0 ≤ i, j <p, the index of tensor product A⊗i ⊗ B⊗j is divisible by p. Then the image ofthe restriction homomorphism K(X)(1) → K(XL)

(1) is divisible by p.

Proof. We have already noticed that since the algebras AL and BL aresplit, the varieties SB(AL) and SB(BL) are isomorphic to ((p−1)-dimensional)projective spaces. Let ξ, η be the Grothendieck classes of the tautologicalline bundles on SB(AL), SB(B)L respectively. The group K(XL) is generatedby {ξiηj}p−1

i,j=0. Using a generalized Peyre’s version [66, Proposition 3.1] ofQuillen’s computation of K-theory for Severi-Brauer schemes [69, Theorem 4.1of §8], one can show that the image of the restriction map K(X) → K(XL) isgenerated by {ind(A⊗i ⊗B⊗j) · ξiηj}p−1

i,j=0.Since the first term of the topological filtration coincides with the kernel

of the rank map, the group K(XL)(1) is generated by {ξiηj − 1}p−1

i,j=0 while the

image of K(X)(1) → K(XL)(1) is generated by {ind(A⊗i⊗B⊗j)·(ξiηj−1)}p−1

i,j=0.The assertion required follows now from the assumption on the indexes

ind(A⊗i ⊗B⊗j).

Corollary 2.4. In the conditions of Lemma 2.3, for any n > 0, theimage of the restriction homomorphism CHn(X) → CHn(XL) is divisible by p.

Proof. For any n ≥ 0, there is a commutative diagram

CHn(XL) −−−→ K(XL)(n/n+1)

resL/F

x xresL/F

CHn(X) −−−→ K(X)(n/n+1)

where the upper arrow is an isomorphism. Therefore, it suffices to show that forany n > 0 the image of the restriction homomorphism K(X)(n) → K(XL)

(n) isdivisible by p. By Lemma 2.3, the image of K(X)(n) → K(XL)

(1) is divisibleby p. Since the quotient K(XL)

(1/n) is torsion-free, we are done.

3. The proof

In this section we prove Theorem 1.1.


We set Fdef= F (T ), A

def= AF , and B

def= BF , where T is the product of

generalized Severi-Brauer varieties written down in the end of Section 1. We

also set Xdef= SB(A)× SB(B) and X

def= SB(A)× SB(B).

First of all we note that by the main property of generalized Severi-Brauervarieties one has ind(A⊗i⊗B) ≤ p for any i = 1, . . . , p−1. Therefore ind(A⊗i⊗B⊗j) ≤ p for any integers i, j ≥ 0, i.e. the algebras A, B are linked. So weonly have to show that they are not strongly linked.

Lemma 3.1. For any not simultaneously 0 integers 0 ≤ i, j < p, the indexof tensor product A⊗i ⊗ B⊗j is divisible by p.

Proof. First of all, by Lemma 2.2, the assertion holds for the algebrasA,B, because they are assumed to be not strongly linked. The assertionon A, B follows now from the index reduction formula for generalized Severi-Brauer varieties ([61, Formula 5.11]) (in fact it suffices to apply a simplerstatement on triviality of the relative Brauer group for the function field ofgeneralized Severi-Brauer varieties).

Let L/F be a common splitting field extension for the algebras A,B. Set

Ldef= L(T ). Clearly, L/F is a common splitting field extension for A, B.

Corollary 3.2. For any n > 0, the image of the restriction homomor-phism CHn(X) → CHn(XL) is divisible by p.

Proof. Follows from Lemma 3.1 and Corollary 2.4.

We consider the graded ring CH∗(X) as a graded CH∗(X)-algebra via therestriction homomorphism CH∗(X) → CH∗(X).

Proposition 3.3. The CH∗(X)-algebra CH∗(X) is generated by its gradedcomponents of codimensions ≤ p.

Proof. Since the pull-back CH∗(X×T ) → CH∗(X) is an epimorphism ofgraded CH∗(X)-algebras (see [39, Theorem 3.1] or Proposition 4.1 of Chapter5 for the surjectivity), it suffices to show that the algebra CH∗(X × T ) isgenerated by its graded components of codimensions ≤ p.

Consider the variety X×T as a scheme over X via the projection. Accord-ing to Proposition 5.3 of Chapter 2, it is a product (over X) of p-grassmanians.By [12, Proposition 14.6.5], CH∗(X × T ) is therefore generated (as CH∗(X)-algebra) by the Chern classes of the tautological bundles on the grassmanians.Since all these bundles have rank p, they may have non-trivial Chern classesonly in codimensions ≤ p.

Finally, Theorem 1.1 follows from Lemma 2.1 and the following assertion:

Corollary 3.4. The image of the restriction CH0(X) → CH0(XL) isdivisible by p2.

3. THE PROOF 161

Proof. Note that since p = 2, we have p < (p − 1)2 = dimX. Thus, byProposition 3.3, the group CH0(X) is generated by the image of CH0(X) →CH0(X) and by the products CHn(X) · CHm(X) with n,m > 0.

The image in CH0(XL) of the first part of the generators is divisible by p2

since in the commutative diagram

CH0(X) −−−→ CH0(XL)x xCH0(X) −−−→ CH0(XL)

the image of the bottom arrow is divisible by p2 (Lemma 2.1).The image in CH0(XL) of the second part of the generators is divisible by

p2 since for any n > 0 the image of CHn(X) → CHn(XL) is divisible by p(Corollary 3.2).


CHAPTER 10

Common splitting fields of division algebras

For every prime number p and every map α : Zn → Z, we find the minimalinteger β such that the following assertion holds: any elements x1, . . . , xn of theBrauer group Br(F ) of an arbitrary field F , satisfying the conditions ind(i1x1+· · ·+ inxn) = pα(i1,...,in) for all i1, . . . , in ∈ Z, possess a finite common splittingfield extension E/F with vp([E : F ]) ≤ β, where vp denotes the multiplicity ofp.

0. Introduction

Let us fix a prime number p. Let α : Zn → Z be an arbitrary map. We saythat α is the behaviour of elements x1, . . . , xn of the Brauer group Br(F ) of afield F , if for any i1, . . . , in ∈ Z the Schur index ind(i1x1 + · · · + inxn) equalspα(i1,...,in). We say that α is a behaviour, if there exists a field F and elementsx1, . . . , xn ∈ Br(F ) with the behaviour α.

Let F be a field and x ∈ Br(F ). A splitting field of x is by definition afield extension E of F such that xE = 0 ∈ Br(E). A common splitting field ofseveral elements of Br(F ) is by definition a field which is a splitting field foreach of the elements. We consider only (common) splitting fields finite overthe base field.

Let us fix a behaviour α. In this chapter we determine the minimal integerβ such that the following assertion holds (see Theorem 3.1): if some elementsin the Brauer group of an arbitrary field F have the behaviour α, then theypossess a common splitting field E with vp([E : F ]) ≤ β.

Similar questions was already considered in the literature. Here is a list ofknown results:1. A classical theorem of Albert (see [1] or [2]) states: if tensor product of twoquaternion division algebras has zero divisors, then the quaternion algebraspossess a common splitting field quadratic over the base field.2. A generalization of Albert’s theorem due to Risman ([70, Theorem 1])asserts: if tensor product of a 2-primary division algebra A and a quaternionalgebra B has zero divisors, then A and B possess a common splitting field ofdegree degA.3. Jacob and Wadsworth ([27], see also Chapter 9) constructed two divisionalgebras of prime degree p over certain field F such that

• ind(A⊗i ⊗F B⊗j) ≤ p for any i, j ≥ 0 and

• the degree of any common splitting field of A and B is divisible by p2.

163

164 10. SPLITTING FIELDS OF ALGEBRAS

4. The following was noticed by M. Rost (unpublished). Three quaternionalgebras such that the Brauer class of any tensor product of some of them isrepresented by a quaternion algebra as well can not be (in general) split by acommon quadratic extension of the base field.5. A generalization of Risman’s theorem to the case of odd prime is obtainedin Chapter 8.

Note that the theorem presented here does not assert existence of a commonsplitting field of degree pβ. We even do not know whether such assertion istrue in general. However, in certain particular cases (this means, for certainconcrete behaviours) the proof can be refined in order to get the stronger result.An example is the theorem of Chapter 8.

Notation. For a smooth variety X over a field F , we denote by K(X) theGrothendieck group of X; by Γ0K(X) the 0-dimensional term of the gamma-filtration on K(X) (for a definition of the gamma-filtration see [52, Definition8.3] or Definition 2.6 of Chapter 1); by T0K(X) the 0-dimensional term ofthe topological filtration on X (see [69, §7] for a definition of the topologicalfiltration). We fix an algebraic closure F of F and denote by X the F -varietyXF .

For any projective homogeneous variety X, we identify K(X) with a sub-group in K(X) via the restriction homomorphism K(X) → K(X) which isinjective by [65].

The order of a finite set S is denoted by |S|.For a central simple F -algebra A, we denote by SB(A) the Severi-Brauer

variety of A and by SB(r, A) (for r ≥ 0) the generalized Severi-Brauer varietiesof A (also called generic partial splitting varieties).

1. “Generic” algebras of given behaviour

For any central simple algebras A1, . . . , An over a field F , we define theirbehaviour to be the behaviour of their classes in the Brauer group of F .

As in Definition 3.5 of Chapter 2, we say that algebras A1, . . . , An aredisjoint, if, for any integers i1, . . . , in ≥ 0, it holds

ind(A⊗i1 ⊗F · · · ⊗F A⊗in) = ind(A⊗i1) · · · ind(A⊗in) .

We say that a collection of algebras A1, . . . , An is “generic” (compare toDefinition 5.4 of Chapter 2), if it can be obtained via the following procedure.We start with some disjoint central simple algebras A1, . . . , An over a field Fsuch that indAj = expAj for every j = 1, . . . , n. Then we take some centralsimple algebras B1, . . . , Bm whose Brauer classes lie in the subgroup of Br(F )generated by the Brauer classes of A1, . . . , An. We denote by Y the directproduct SB(r1, A1) × · · · × SB(rm, Am) of generalized Severi-Brauer varieties

with some r1, . . . , rm ≥ 0 and we set Aidef= (Ai)F (Y ) for each i = 1, . . . , n.

1. “GENERIC” ALGEBRAS OF GIVEN BEHAVIOUR 165

Proposition 1.1. For any behaviour α : Zn → Z, there exist “generic”division algebras A1, . . . , An (over a suitable field F ) having the behaviour α.

We prove the proposition after the following

Lemma 1.2. Let A,B be central simple algebras over a field F and letA′, B′ be central simple algebras over a field F ′. Suppose that degB = degB′

and that for any i ≥ 0 the index of A′ ⊗F ′ B′⊗i divides the index of A⊗F B⊗i.

Then, for any r ≥ 0, the index of A′F ′(SB(r,B′)) divides the index of AF (SB(r,B)).

Proof. Set sdef= degB = degB′ and denote by d the greatest common

divisor of r and s. By [60, Formula 1], one has

ind(AF (SB(r,B))) = gcd1≤i≤s

(d

gcd(i, d)ind(A⊗B⊗j)

).

Replacing A by A′ and B by B′, we get a formula for ind(A′F ′(SB(r,B′))). Since

ind(A′ ⊗B′⊗i) divides ind(A⊗B⊗i) for any i, we are done.

Proof of Proposition 1.1. We start with disjoint division algebras A1,. . . , An over a suitable field F such that for any j = 1, . . . , n one has

degAj = expAj = pα(0,...,0,1,0,...,0)

where 1 (in the argument of α) is placed on the j-th position (algebras likethat do definitely exist). For every i1, . . . , in with 0 ≤ ij < degAj, we considerthe algebra

Bi1...indef= A⊗i1

1 ⊗ · · · ⊗ A⊗inn

and denote by Yi1...in the variety SB(pα(i1,...,in), Bi1...in). We set Ydef=∏Yi1...in

and Ajdef= (Aj)F (Y ) for all j = 1, . . . , n. We state A1, . . . , An are “generic”

division algebras required.

To show that ind(i1[A1] + · · ·+ in[An]) = pα(i1,...,in)

for all i1, . . . , in ∈ Z, itsuffices to check that

ind(A⊗i11 ⊗ · · · ⊗ A⊗in

n ) = pα(i1,...,in)

for any i1, . . . , in with 0 ≤ ij < degAj. Since the inequality ≤ is evident, itsuffices to prove the inverse inequality.

Since α is a behaviour, we can find division algebras A′1, . . . , A

′n over a

field F ′ having the behaviour α. Clearly, for any i1, . . . , in with 0 ≤ ij <

degA′j = degAj, the index of the algebra B′

i1...in

def= A′⊗i1

1 ⊗ · · · ⊗ A′⊗inn equals

pα(i1,...,in) and divides the index of the algebra Bi1...in (while their degrees co-incide). Let us choose some integers i′1, . . . , i

′n with 0 ≤ i′j ≤ degA′

j. By

Lemma 1.2, ind(B′i1...in

)F ′(Y ′i′1...i

′n) divides ind(Bi1...in)F (Yi′1...i

′n), where Y

′i′1...i

′n

def=

SB(pα(i′1,...,i

′n), B′

i′1...i′n). Moreover, the extension F ′(Y ′

i′1...i′n)/F does not in fact

affect the index of any F ′-algebra, because the variety Y ′i′1...i

′nis rational. There-

fore, the index of B′i1...in

itself divides ind(Bi1...in)F (Yi′1...i′n).


So we see, that if we replace the base field F of the algebras A1, . . . , Anby the function field F (Yi′1...i′n), the index of every Bi1...in is still divisible by

pα(i1,...,in).Passing after that to the function field of another variety Yi′′1 ...i′′n and so on,

we get in the end the required statement on the indexes.

2. Definition of β

We fix a prime number p and a behaviour α : Zn → Z.Let us consider a field F and elements x1, . . . , xn ∈ Br(F ) with the be-

haviour α. Choose division F -algebras representing the elements x1, . . . , xn

and denote by X1, . . . , Xn the corresponding Severi-Brauer varieties. Set Xdef=

X1 × · · · ×Xn.Since X is isomorphic to a direct product of projective spaces, Γ0K(X) is

an infinite cyclic group generated by the class of a rational point. We have0 = Γ0K(X) ⊂ Γ0K(X) ≃ Z. Therefore, the quotient Γ0K(X)/Γ0K(X) is afinite group.

Definition 2.1. We put βdef= vp(|Γ0K(X)/Γ0K(X)|).

Lemma 2.2. The integer β, defined in 2.1, depends only on the prime pand the behaviour α; it does not depend on the choice of the field F and theelements x1, . . . , xn ∈ Br(F ).

Proof. According to Corollary 2.2 of Chapter 2, the groups Γ0K(X) andΓ0K(X) depend only on p and on the behaviour of x1, . . . , xn.

3. The theorem

Theorem 3.1. For a prime number p and a behaviour α : Zn → Z, let βbe the integer defined in the previous section. Then

1. for any field F , any n elements x1, . . . , xn ∈ Br(F ) with the behaviourα possess a common splitting field E/F satisfying the condition

vp([E : F ]) ≤ β;

2. there exists a field F and elements x1, . . . , xn ∈ Br(F ) with the behaviourα such that any their common splitting field E/F satisfies the condition

vp([E : F ]) ≥ β .

We prove the theorem after the following

Lemma 3.2. Let A1, . . . , An be central simple algebras over a field F and

let X1, . . . , Xn be the corresponding Severi-Brauer varieties. Set Xdef= X1 ×

· · · ×Xn and β′ def= vp(|T0K(X)/T0K(X)|). Then

1. for any common splitting field E/F of A1, . . . , An, it holds

vp([E : F ]) ≥ β′;

3. THE THEOREM 167

2. the algebras A1, . . . , An possess a common splitting field E/F with

vp([E : F ]) = β′ .

Proof. For any variety Y , T0K(Y ) is by definition the subgroup of K(Y )generated by the classes [y] ∈ K(Y ) of the closed points y ∈ Y . If Y is a

complete F -variety, the rule [y] 7→ deg(y)def= [F (y) : F ], where F (y) is the

residue field of y, determines a well-defined homomorphism deg : T0K(Y ) → Z(compare to [14, Corollary 6.10 of Chapter II]). Note that the compositionT0K(Y ) → T0K(Y ) → Z of the restriction homomorphism with the degreehomomorphism for Y coincides with the degree homomorphism for Y .

Since X is isomorphic to a direct product of projective spaces, the homo-morphism deg : T0K(X) → Z is bijective. In particular, since T0K(X) is anon-zero subgroup of T0K(X), we see that the quotient T0K(X)/T0K(X) isfinite.

1. If E is a common splitting field of the algebras A1, . . . , An, the varietyXE has a closed rational point. Therefore, there exists a zero-cycle on X ofdegree [E : F ]. It follows that the order of the quotient T0K(X)/T0K(X)divides [E : F ]. In particular, vp([E : F ]) ≥ β′.

2. It follows from the definition of β′ and the above discussion that thereexists a zero-cycle σ =

∑ri=1 liσi on X (where li ∈ Z and σi ∈ X) with

vp(deg(σ)) = β′. Since

deg(σ)def=

r∑i=1

li deg(σi),

one has vp(deg(σi)) ≤ β′ for certain i. Denote by E the residue field of the pointσi. Since the variety XE possess a rational point, E is a common splitting fieldof the algebras A1, . . . , An. Therefore, by Item 1, it holds vp([E : F ]) ≥ β′.From the other hand vp([E : F ]) = vp(deg(σi)) ≤ β′. Thus vp([E : F ]) =β′.

Proof of Theorem 3.1. 1. Let x1, . . . , xn be some elements with thebehaviour α in the Brauer group of a field F . Consider the varietyX as in Defi-nition 2.1. According to Item 2 of Lemma 3.2, the elements x1, . . . , xn possess a

common splitting field E/F with vp([E : F ]) = β′ def= vp(|T0K(X)/T0K(X)|).

From the other hand, β = vp(Γ0K(X)/Γ0K(X)) by Lemma 2.2. Since

T0K(X) = Γ0K(X) and T0K(X) ⊃ Γ0K(X)

(see [13, Theorem 3.9 of Chapter V] for the second relation) the order of thequotient T0K(X)/T0K(X) divides the order of the quotient Γ0K(X)/Γ0K(X).Therefore β′ ≤ β and consequently vp([E : F ]) ≤ β.

2. Let x1, . . . , xn be the Brauer classes of some “generic” division algebraswith the behaviour α (which exist by Proposition 1.1). LetX be the product ofthe Severi-Brauer varieties of these division algebras. By Item 1 of Lemma 3.2,vp([E : F ]) ≥ β′ for any common splitting field E/F of x1, . . . , xn. By Theorem5.5 of Chapter 2, one has T0K(X) = Γ0K(X). Therefore β′ = β.


Bibliography

[1] Albert, A. A. On the Wedderburn norm condition for cyclic algebras. Bull. Amer. Math.Soc. 37 (1931), 301–312.

[2] Tensor product of quaternion algebras. Proc. Amer. Math. Soc. 35 (1972), 65–66.[3] Structure of Algebras. Amer. Math. Soc. Coll. Pub. XXIV, Providence R. I. 1961.[4] Amitsur, S. A., Rowen L. H., Tignol J.-P. Division algebras of degree 4 and 8 with

involution. Israel J. Math. 33 (1979), 133–148.[5] Arason, J. Kr. Cohomologische Invarianten quadratischer Formen. J. Algebra 36

(1975), 448–491.[6] Artin, M. Brauer-Severi varieties. In Brauer Groups in Ring Theory and Algebraic

Geometry, edited by F. van Oystaeyen and A. Verschoren. Springer Lect. Notes Math.917 (1982), 194–210.

[7] Blanchet, A. Function fields of generalized Brauer-Severi varieties. Comm. Algebra 19(1991), no. 1, 97–118.

[8] Demazure, M., Gabriel, P. Groupes Algebriques (Tome I). Masson & Cie, Editeur —Paris, North-Holland Publishing Company — Amsterdam, 1970.

[9] Elman, R., Lam, T. Y. Pfister forms and K-theory of fields. J. of Algebra 23 (1972),181–213.

[10] Esnault, H., Kahn, B., Levine M., and Viehweg V. The Arason invariant and mod 2algebraic cycles. K-theory Preprint Archives (http://www.math.uiuc.edu/K-theory/),Preprint N◦151.

[11] Faith, C. Algebra: Rings, Modules and Categories I. Springer-Verlag , 1973.[12] Fulton, W. Intersection Theory. Springer-Verlag, 1984.[13] , Lang, S. Riemann-Roch Algebra. Springer-Verlag, 1985.[14] Hartshorne, R. Algebraic Geometry. Springer-Verlag, 1977.[15] Hoffmann, D. W. Isotropy of 5-dimensional quadratic forms over the function field of

a quadric. Proc. Symp. Pure Math. 58.2 (1995), 217-225.[16] On 6-dimensional quadratic forms isotropic over the function field of a quadric.

Comm. Alg. 22 (1994), 1999–2014.[17] Twisted Pfister Forms. Doc. Math. 1 (1996), 67–102.[18] On the dimensions of anisotropic quadratic forms in I4. Invent. Math., to ap-

pear.[19] Splitting patterns and invariants of quadratic forms. Math. Nachr., to appear.[20] , Lewis, D. W., van Geel, J. Minimal forms for function fields of conics. Proc.

Symp. Pure Math. 58.2 (1995), 227–237.[21] , Tignol, J.-P. On 14-dimensional forms in I3, 8-dimensional forms in I2, and

the common value property. Preprint, 1997.[22] Hurrelbrink, J., Rehmann, U. Splitting patterns of quadratic forms. Math. Nachr. 176

(1995), 111–127.[23] Izhboldin O. T.On the Nonexcellence of Field Extensions F (π)/F . Doc. Math. 1 (1996),

127–136.

169

170 BIBLIOGRAPHY

[24] On the nonexcellence of the function fields of Severi-Brauer varieties. Max-Planck-Institut fur Mathematik in Bonn, Preprint MPI 96-159 (1996), 1–28.

[25] Isotropy of low dimensional forms over the function fields of quadrics. Max-Planck-Institut fur Mathematik in Bonn, Preprint MPI 97-1 (1997), 1–18.

[26] Jacob, B. Indecomposible division algebras of prime exponent. J. reine angew. Math.413 (1991), 181–197.

[27] , Wadsworth, A. R. Division algebras with no common subfields. Israel J. Math.83 (1993), 353–360.

[28] Jacobson, N. Some applications of Jordan norms to involutorial associative algebras.Adv. Math. 48 (1983), 149–165.

[29] Kahn, B. Descente galoisienne et K2 des corps de nombres. K-Theory 7 (1993), 55–100.[30] Quadratic forms isotropic over the function field of a quadric: a survey. Math.

Forschungsinstitut Oberwolfach, Tagungsbericht 24/1995 (1995), 6–7.[31] Karpenko, N. A. Chow ring of a projective quadric. Ph. D. theses (in Russian),

Leningrad (1990), 80 p.[32] Algebro-geometric invariants of quadratic forms. Algebra i Analiz 2 (1991), no.

1, 141–162 (in Russian). Engl. transl.: Leningrad (St. Petersburg) Math. J. 2 (1991),no. 1, 119–138.

[33] On topological filtration for Severi-Brauer varieties. Proc. Symp. Pure Math.58.2 (1995), 275–277.

[34] On topological filtration for Severi-Brauer varieties II. Amer. Math. Soc. Transl.174 (1996), no. 2, 45–48.

[35] Torsion in CH2 of Severi-Brauer varieties and indecomposability of generic al-gebras. Manuscripta Math. 88 (1995), 109–117.

[36] On topological filtration for triquaternion algebra. Nova J. of Math., Game The-ory, and Algebra 4 (1996), no. 2, 161–167.

[37] Grothendieck’s Chow motives of Severi-Brauer varieties. Algebra i Analiz 7(1995), no.4, 196–213 (in Russian). Engl. transl.: St. Petersburg Math. J. 7 (1996),no. 4, 649–661.

[38] Cohomology of relative cellular spaces and isotropic flag varieties. Preprint(1997), 1–57.

[39] , Merkurjev, A. S. Chow groups of projective quadrics. Algebra i Analiz 2 (1990),no. 3, 218–235 (in Russian). Engl. transl.: Leningrad (St. Petersburg) Math. J. 2 (1991),no. 3, 655–671.

[40] Knebusch, M. Generic splitting of quadratic forms I. Proc. London Math. Soc. 33(1976), 65–93.

[41] Generic splitting of quadratic forms II. Proc. London Math. Soc. 34 (1977),1–31.

[42] Knus, M.-A., Merkurjev, A. S., Rost, M., Tignol, J.-P. The Book of Involutions. Pre-liminary version of June 12, 1996.

[43] Kock, B. Chow motif and higher Chow theory of G/P . Manuscripta Math. 70 (1991),363–372.

[44] Laghribi, A. Formes quadratiques de dimension 6. Preprint (1996).[45] Isotropie d’une forme quadratique de dimension ≤ 8 sur le corps des fonctions

d’une quadrique. C.R.Acad.Sci.Paris 323 (1996), serie I, 495–499.[46] Isotropie de certaines formes quadratiques de dimension 7 et 8 sur le corps des

fonctions d’une quadrique. Duke Math. J. 85 (1996), no. 2, 397–410.[47] Formes quadratiques en 8 variables dont l’algebre de Clifford est d’indice 8. K-

Theory, to appear.[48] Lam, T. Y. The Algebraic Theory of Quadratic Forms. Massachusetts: Benjamin 1973

(revised printing: 1980).[49] Leep, D. Function fields results. Handwritten notes (1989).

BIBLIOGRAPHY 171

[50] Levine, M., Srinivas, V., Weyman, J. K-theory of twisted Grassmanians. K-Theory 3(1989) 99–121.

[51] Mammone, P. On the tensor product of division algebras. Arch. Math. 58 (1992), 34–39.[52] Manin, Yu. I. Lectures on the K-functor in algebraic geometry. Russian Math. Surveys

24 (1969), no. 5, 1–89.[53] Merkurjev, A. S. On the norm residue symbol of degree 2. Dokl. Akad. Nauk SSSR 261

(1981), 542–547 (in Russian). Engl. transl.: Soviet Math. Dokl. 24 (1981), 546–551.[54] Kaplansky conjecture in the theory of quadratic forms. Zap. Nauchn. Semin.

Leningr. Otd. Mat. Inst. Steklova 175 (1989), 75–89 (in Russian). Engl. transl.: J. Sov.Math. 57 (1991), no. 6, 3489–3497.

[55] Simple algebras and quadratic forms. Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991),218–224 (in Russian). English transl.: Math. USSR Izv. 38 (1992), no. 1, 215–221.

[56] K-theory of simple algebras. Proc. Symp. Pure Math. 58.1 (1995), 65–83.[57] The group H1(X,K2) for projective homogeneous varieties. Algebra i Analiz 7

(1995), no. 3, 136–164 (in Russian). Engl. transl.: St. Petersburg Math. J. 7 (1996),no. 3, 421–444.

[58] Certain K-cohomology groups of Severi-Brauer varieties. Proc. Symp. PureMath. 58.2 (1995), 319–331.

[59] , Panin, I. A. K-theory of algebraic tori and toric varieties. K-Theory (1997), toappear.

[60] , Panin I. A., Wadsworth, A. A list of index reduction formulas. Preprint 94-079,SFB preprint series, Bielefeld Univ., Germany, 1994.

[61] , Panin, I. A., Wadsworth, A. R. Index reduction formulas for twisted flag vari-eties, I. K-Theory 10 (1996), 517–596.

[62] , Suslin, A. A. The group K3 for a field. Izv. Akad. Nauk SSSR Ser. Mat. 54(1990), no.3, 522–545 (in Russian). Engl. transl.: Math. USSR, Izv. 36 (1991), no.3,541–565.

[63] Mumford, D. Introduction to Algebraic Geometry (Preliminary version of first 3 Chap-ters).

[64] Panin, I. A. K-theory of Grassman bundles and it’s twisted forms. LOMI, preprintE-5-89, Leningrad 1989.

[65] On the algebraic K-theory of twisted flag varieties. K-Theory 8 (1994), no. 6,541–585.

[66] Peyre, E. Products of Severi-Brauer varieties and Galois cohomology. Proc. Symp. PureMath. 58.2 (1995), 369–401.

[67] Corps de fonctions de varietes homogenes et cohomologie galoisienne. C. R.Acad. Sci. Paris 321 (1995), serie I, 891–896.

[68] Pfister, A. Quadratische Formen in beliebigen Korpern. Invent. Math. 1 (1966), 116–132.

[69] Quillen, D. Higher algebraic K-theory: I. In Algebraic K-Theory I, edited by H. Bass.Springer Lect. Notes Math. 341 (1973), 85–147.

[70] Risman, L. J. Zero divisors in tensor products of division algebras. Proc. Amer. Math.Soc. 51 (1975), no. 1, 35–36.

[71] Rost, M. Hilbert 90 for K3 for degree-two extensions. Preprint (1986).[72] On 14-dimensional quadratic forms, their spinors and the difference of two oc-

tonion algebras. Preprint (1994).[73] Rowen, L. H. Division algebras of exponent 2 and characteristic 2. J. of Algebra 90

(1984), 71–83.[74] Sansuc, J.-J. Groupe de Brauer et arithmetique des groupes algebriques lineaires sur un

corps de nombres. J. reine angew. Math. 327 (1981), 12–80.[75] Schapiro D. B. Similarities, quadratic forms. and Clifford algebra. Doctoral Disserta-

tion, University of California, Berkeley, California (1974).

172 BIBLIOGRAPHY

[76] Scharlau, W. Quadratic and Hermitian Forms Springer, Berlin, Heidelberg, New York,Tokyo (1985).

[77] Scheiderer, C. Real and etale cohomology. Lect. Notes Math. 1588 (1994).[78] Schofield, A., van den Bergh, M. The index of a Brauer class on a Brauer-Severi variety.

Transactions Amer. Math. Soc. 333 (1992), no. 2, 729–739.[79] Serre, J.-P. Cohomologie galoisienne. Lect. Notes Math. 5 (1994) (5-eme edition).[80] Srinivas, V. Algebraic K-Theory. Progress in Math. 90, Birkhauser 1991.[81] Suslin, A. A. Algebraic K-theory and the norm residue homomorphism. J. Soviet Math.

30 (1985), 2556–2611.[82] Swan, R. K-theory of quadric hypersurfaces. Ann. Math. 122 (1985), no. 1, 113–154.[83] Tignol, J.-P. Algebres indecomposibles d’exposant premier. Adv. in Math. 63 (1987),

205–228.[84] Reduction de l’indice d’une algebre simple centrale sur le corps des fonctions

d’une quadrique. Bull. Soc. Math. Belgique 42 (1990), 725–745.[85] , Wadsworth, A. R. Totally ramified valuations on finite-dimensional division

algebras. Trans. Amer. Math. Soc. 302 (1987), 223–250.[86] Wadsworth, A. R. Similarity of quadratic forms and isomorphism of their function

fields. Trans. Amer. Math. Soc. 208 (1975), 352–358.

Documents

Cohomological Invariants of Homogeneous Varietieskarpenko/publ/hbl.pdf · Galois cohomology group H3(F(X)/F,Z/2), where F(X) is the function ﬁeld of X. We compute these groups for